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<!--XBRL Document Created with the Workiva Platform-->
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  <link:labelLink xlink:role="http://www.xbrl.org/2003/role/link" xlink:type="extended">
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    <link:label id="lab_us-gaap_InvestmentPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_InvestmentPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment, Policy [Policy Text Block]</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsLineItems_label_en-US" xlink:label="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt and Equity Securities, FV-NI [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsLineItems"/>
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    <link:label id="lab_kg_ScheduleOfPriorYearLossDevelopmentByLineOfBusinessTableTextBlock_documentation_en-US" xlink:label="lab_kg_ScheduleOfPriorYearLossDevelopmentByLineOfBusinessTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Schedule of prior year loss development by line of business</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ScheduleOfPriorYearLossDevelopmentByLineOfBusinessTableTextBlock" xlink:href="kg-20250930.xsd#kg_ScheduleOfPriorYearLossDevelopmentByLineOfBusinessTableTextBlock"/>
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    <link:label id="lab_ecd_AwardTmgPredtrmndFlag_label_en-US" xlink:label="lab_ecd_AwardTmgPredtrmndFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Timing Predetermined [Flag]</link:label>
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    <link:label id="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Commission Rate, Adjustment Criteria, Loss Ratio Floor</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Commission Rate, Adjustment Criteria, Loss Ratio Floor</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor" xlink:to="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioFloor" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdditionalPaidInCapitalMember_6bc582a5-3c3b-447d-a528-ac300b753358_terseLabel_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapitalMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional paid-in capital</link:label>
    <link:label id="lab_us-gaap_AdditionalPaidInCapitalMember_label_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapitalMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Additional Paid-in Capital [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdditionalPaidInCapitalMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdditionalPaidInCapitalMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdditionalPaidInCapitalMember" xlink:to="lab_us-gaap_AdditionalPaidInCapitalMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings_789c2128-f19e-401d-9077-ddbf158ecf59_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Quarterly brokerage fee, above portfolio threshold, percent of average holdings</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Quarterly Brokerage Fee, Above Portfolio Threshold, Percent of Average Holdings</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Quarterly Brokerage Fee, Above Portfolio Threshold, Percent of Average Holdings</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings" xlink:to="lab_kg_RelatedPartyTransactionQuarterlyBrokerageFeeAbovePortfolioThresholdPercentOfAverageHoldings" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TabularListTableTextBlock_terseLabel_en-US" xlink:label="lab_ecd_TabularListTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Tabular List, Table</link:label>
    <link:label id="lab_ecd_TabularListTableTextBlock_label_en-US" xlink:label="lab_ecd_TabularListTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Tabular List [Table Text Block]</link:label>
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    <link:label id="lab_kg_EarnOutLiabilityPolicyTextBlock_2ae615fc-05de-4165-879b-dde0058e04cb_terseLabel_en-US" xlink:label="lab_kg_EarnOutLiabilityPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Earn Out Liability</link:label>
    <link:label id="lab_kg_EarnOutLiabilityPolicyTextBlock_label_en-US" xlink:label="lab_kg_EarnOutLiabilityPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earn Out Liability [Policy Text Block]</link:label>
    <link:label id="lab_kg_EarnOutLiabilityPolicyTextBlock_documentation_en-US" xlink:label="lab_kg_EarnOutLiabilityPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Earn Out Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EarnOutLiabilityPolicyTextBlock" xlink:href="kg-20250930.xsd#kg_EarnOutLiabilityPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EarnOutLiabilityPolicyTextBlock" xlink:to="lab_kg_EarnOutLiabilityPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_1e840011-3956-4ec9-a89d-28fef4739434_terseLabel_en-US" xlink:label="lab_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Recently Adopted Accounting Standards and Recently Issued Accounting Standards Not Yet Adopted</link:label>
    <link:label id="lab_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">New Accounting Pronouncements, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:to="lab_us-gaap_NewAccountingPronouncementsPolicyPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock_ef6f14cb-1cc2-4682-9882-945225190b48_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Quota Share Arrangements with AmTrust</link:label>
    <link:label id="lab_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Related Party Transactions [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock" xlink:to="lab_us-gaap_ScheduleOfRelatedPartyTransactionsTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_DiversifiedLegacySegmentMember_d6b5c70a-7285-4aa1-b2ab-4cf29beb9a1b_verboseLabel_en-US" xlink:label="lab_kg_DiversifiedLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Diversified Legacy Reinsurance</link:label>
    <link:label id="lab_kg_DiversifiedLegacySegmentMember_label_en-US" xlink:label="lab_kg_DiversifiedLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Diversified Legacy Segment [Member]</link:label>
    <link:label id="lab_kg_DiversifiedLegacySegmentMember_documentation_en-US" xlink:label="lab_kg_DiversifiedLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Diversified Legacy Segment</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_DiversifiedLegacySegmentMember" xlink:href="kg-20250930.xsd#kg_DiversifiedLegacySegmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_DiversifiedLegacySegmentMember" xlink:to="lab_kg_DiversifiedLegacySegmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueByBalanceSheetGroupingTextBlock_fb9d339d-62f2-4068-ae75-637de961637e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByBalanceSheetGroupingTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Carrying Value and Fair Value of Senior Notes</link:label>
    <link:label id="lab_us-gaap_FairValueByBalanceSheetGroupingTextBlock_label_en-US" xlink:label="lab_us-gaap_FairValueByBalanceSheetGroupingTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, by Balance Sheet Grouping [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueByBalanceSheetGroupingTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueByBalanceSheetGroupingTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueByBalanceSheetGroupingTextBlock" xlink:to="lab_us-gaap_FairValueByBalanceSheetGroupingTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInUnearnedPremiumsNet_f8572d5d-e830-453a-8178-d82b003e493e_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInUnearnedPremiumsNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in unearned premiums</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInUnearnedPremiumsNet_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInUnearnedPremiumsNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">(Increase) Decrease in Unearned Premiums, Net</link:label>
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    <link:label id="lab_us-gaap_FairValueByMeasurementBasisAxis_85dd351b-1792-457d-b8d9-6d2fe1179ef1_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByMeasurementBasisAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measurement Basis [Axis]</link:label>
    <link:label id="lab_us-gaap_FairValueByMeasurementBasisAxis_label_en-US" xlink:label="lab_us-gaap_FairValueByMeasurementBasisAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Basis [Axis]</link:label>
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    <link:label id="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock_11916e8e-d422-4304-bb3a-dfa225496a60_verboseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transactions</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transactions Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:to="lab_us-gaap_RelatedPartyTransactionsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NamedExecutiveOfficersFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Named Executive Officers, Footnote</link:label>
    <link:label id="lab_ecd_NamedExecutiveOfficersFnTextBlock_label_en-US" xlink:label="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Named Executive Officers, Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NamedExecutiveOfficersFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NamedExecutiveOfficersFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NamedExecutiveOfficersFnTextBlock" xlink:to="lab_ecd_NamedExecutiveOfficersFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability_a52ab466-80d9-48f7-bcc9-a2d89ddb781f_negatedTerseLabel_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value of earn out liability</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability_127870f3-e10f-46f6-8214-62b223671f9b_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in fair value of earn out liability</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability_5add9690-2a73-430d-8742-8bb07b185a01_verboseLabel_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Decrease in fair value of earn out liability</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability_label_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Contingent Consideration Arrangements, Change in Amount of Contingent Earn-Out, Liability</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability_documentation_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Contingent Consideration Arrangements, Change in Amount of Contingent Earn-Out Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:href="kg-20250930.xsd#kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:to="lab_kg_BusinessCombinationContingentConsiderationArrangementsChangeInAmountOfContingentEarnOutLiability" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty_d14438bc-9a5f-448b-bcd8-f8f05bb335ee_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percent of premiums written, net of cost of unaffiliated inuring reinsurance from related party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Premium Written, Net of Cost of Unaffiliated Inuring Reinsurance From Related Party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Premium Written, Net of Cost of Unaffiliated Inuring Reinsurance From Related Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:to="lab_kg_RelatedPartyTransactionPercentOfPremiumWrittenNetOfCostOfUnaffiliatedInuringReinsuranceFromRelatedParty" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedMaturitiesAmortizedCostBasis_8a3f31c1-2e4c-4baa-8cbe-06ab258ca4e5_terseLabel_en-US" xlink:label="lab_kg_FixedMaturitiesAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortized cost</link:label>
    <link:label id="lab_kg_FixedMaturitiesAmortizedCostBasis_label_en-US" xlink:label="lab_kg_FixedMaturitiesAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Amortized Cost Basis</link:label>
    <link:label id="lab_kg_FixedMaturitiesAmortizedCostBasis_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Amortized Cost Basis</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesAmortizedCostBasis" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesAmortizedCostBasis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesAmortizedCostBasis" xlink:to="lab_kg_FixedMaturitiesAmortizedCostBasis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfStockByClassTable_7e36f10b-dbfb-4bf9-93db-d633f14b7667_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfStockByClassTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Stock by Class [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfStockByClassTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfStockByClassTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock, Class of Stock [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfStockByClassTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfStockByClassTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfStockByClassTable" xlink:to="lab_us-gaap_ScheduleOfStockByClassTable" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee_4d82b95d-c810-40f1-8c34-62d85f58edc5_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percent of ceded underwriting risk in exchange for ceding fee</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Ceded Underwriting Risk in Exchange for Ceding Fee</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Ceded Underwriting Risk in Exchange for Ceding Fee</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:to="lab_kg_RelatedPartyTransactionPercentOfCededUnderwritingRiskInExchangeForCedingFee" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AmTrustQuotaShareMember_9181af84-f74f-4134-9e4f-65b9609e0d0e_terseLabel_en-US" xlink:label="lab_kg_AmTrustQuotaShareMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">AmTrust Quota Share</link:label>
    <link:label id="lab_kg_AmTrustQuotaShareMember_label_en-US" xlink:label="lab_kg_AmTrustQuotaShareMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AmTrust Quota Share [Member]</link:label>
    <link:label id="lab_kg_AmTrustQuotaShareMember_documentation_en-US" xlink:label="lab_kg_AmTrustQuotaShareMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">AmTrust Quota Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AmTrustQuotaShareMember" xlink:href="kg-20250930.xsd#kg_AmTrustQuotaShareMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_AmTrustQuotaShareMember" xlink:to="lab_kg_AmTrustQuotaShareMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold_83122f28-5f5d-4bce-8a87-4001fa958458_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party transaction, loan agreement, obligation threshold</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Loan Agreement, Obligation Threshold</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Loan Agreement, Obligation Threshold</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionLoanAgreementObligationThreshold"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:to="lab_kg_RelatedPartyTransactionLoanAgreementObligationThreshold" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation_05b1931a-fc80-4aab-a42b-e9a8ea915608_terseLabel_en-US" xlink:label="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Assets held for sale</link:label>
    <link:label id="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation_56963b65-cd57-4cfb-bbc9-78c801f9a754_totalLabel_en-US" xlink:label="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Assets held for sale</link:label>
    <link:label id="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation_label_en-US" xlink:label="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:to="lab_us-gaap_AssetsOfDisposalGroupIncludingDiscontinuedOperation" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AdjToCompAmt_terseLabel_en-US" xlink:label="lab_ecd_AdjToCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment to Compensation, Amount</link:label>
    <link:label id="lab_ecd_AdjToCompAmt_label_en-US" xlink:label="lab_ecd_AdjToCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AdjToCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AdjToCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AdjToCompAmt" xlink:to="lab_ecd_AdjToCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet_f7c5ca9f-1507-45c3-becd-4fa9313a954b_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Less: unamortized fair value adjustment</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Unamortized Discount (Premium) and Debt Issuance Costs, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet" xlink:to="lab_us-gaap_DebtInstrumentUnamortizedDiscountPremiumAndDebtIssuanceCostsNet" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets_385b8cfe-9f15-426a-9b04-b40dde156407_negatedLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Other assets</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Other Operating Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOtherOperatingAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:to="lab_us-gaap_IncreaseDecreaseInOtherOperatingAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ConsolidationPolicyTextBlock_91347eda-d603-441b-aa0d-1ad35b05b41e_terseLabel_en-US" xlink:label="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basis of Reporting and Consolidation</link:label>
    <link:label id="lab_us-gaap_ConsolidationPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Consolidation, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ConsolidationPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ConsolidationPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ConsolidationPolicyTextBlock" xlink:to="lab_us-gaap_ConsolidationPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EmployeeStockOptionMember_terseLabel_en-US" xlink:label="lab_us-gaap_EmployeeStockOptionMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Employee Stock Option</link:label>
    <link:label id="lab_us-gaap_EmployeeStockOptionMember_label_en-US" xlink:label="lab_us-gaap_EmployeeStockOptionMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement, Option [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EmployeeStockOptionMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EmployeeStockOptionMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EmployeeStockOptionMember" xlink:to="lab_us-gaap_EmployeeStockOptionMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock_b4831b63-0d52-45e1-999a-8a7875279578_terseLabel_en-US" xlink:label="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Amortized Cost, Gross Unrealized Gains and Losses and Fair Value on Fixed Maturities</link:label>
    <link:label id="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock_label_en-US" xlink:label="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Investments Reconciliation [Table Text Block]</link:label>
    <link:label id="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock_documentation_en-US" xlink:label="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Schedule of Investments Reconciliation [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:href="kg-20250930.xsd#kg_ScheduleOfInvestmentsReconciliationTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:to="lab_kg_ScheduleOfInvestmentsReconciliationTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsPolicyTextBlock_4565d23b-0706-4c18-81ec-1864d4a5127a_terseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalents</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalents, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndCashEquivalentsPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:to="lab_us-gaap_CashAndCashEquivalentsPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DepositContractsLiabilities_6f663575-66e8-4dc6-a923-2f25c9677d71_terseLabel_en-US" xlink:label="lab_us-gaap_DepositContractsLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Deposit contracts</link:label>
    <link:label id="lab_us-gaap_DepositContractsLiabilities_label_en-US" xlink:label="lab_us-gaap_DepositContractsLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deposit Contracts, Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DepositContractsLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DepositContractsLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DepositContractsLiabilities" xlink:to="lab_us-gaap_DepositContractsLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DividendsCommonStockCash_6071c31c-2b28-4c54-9c35-d23baed40414_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_DividendsCommonStockCash" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Distribution to shareholders</link:label>
    <link:label id="lab_us-gaap_DividendsCommonStockCash_label_en-US" xlink:label="lab_us-gaap_DividendsCommonStockCash" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Dividends, Common Stock, Cash</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DividendsCommonStockCash" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DividendsCommonStockCash"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DividendsCommonStockCash" xlink:to="lab_us-gaap_DividendsCommonStockCash" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockholdersEquity_d56509df-9562-41b9-9cb7-3e09453c84df_totalLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total shareholders&#8217; equity</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_fcd8556b-6db5-408a-a46f-1690a27726cd_periodStartLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Beginning balance</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_4715e18e-bdf5-4963-a2bc-9b771f3fb6de_periodEndLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Ending balance</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_f3f95c6b-c2a1-49c5-b10d-452775c578bf_terseLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cumulative translation adjustment</link:label>
    <link:label id="lab_us-gaap_StockholdersEquity_label_en-US" xlink:label="lab_us-gaap_StockholdersEquity" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity, Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquity" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquity"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockholdersEquity" xlink:to="lab_us-gaap_StockholdersEquity" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome_fd5dd4de-49db-446b-a280-1907902b77c8_terseLabel_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued investment income</link:label>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome_label_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Accrued Investment Income</link:label>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome_documentation_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Accrued Investment Income</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:href="kg-20250930.xsd#kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:to="lab_kg_DisposalGroupIncludingDiscontinuedOperationAccruedInvestmentIncome" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueByMeasurementFrequencyAxis_e9dcdc0e-d5cc-4472-9de3-e6e54bd6638e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByMeasurementFrequencyAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measurement Frequency [Axis]</link:label>
    <link:label id="lab_us-gaap_FairValueByMeasurementFrequencyAxis_label_en-US" xlink:label="lab_us-gaap_FairValueByMeasurementFrequencyAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Frequency [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueByMeasurementFrequencyAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueByMeasurementFrequencyAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueByMeasurementFrequencyAxis" xlink:to="lab_us-gaap_FairValueByMeasurementFrequencyAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleTable_d5d31cdd-da3c-4b40-9111-39d8ff5f821c_terseLabel_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Available-for-sale Securities [Table]</link:label>
    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleTable_label_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleTable" xlink:to="lab_us-gaap_DebtSecuritiesAvailableForSaleTable" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_InsiderTradingPoliciesProcLineItems_label_en-US" xlink:label="lab_ecd_InsiderTradingPoliciesProcLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_InsiderTradingPoliciesProcLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_InsiderTradingPoliciesProcLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_InsiderTradingPoliciesProcLineItems" xlink:to="lab_ecd_InsiderTradingPoliciesProcLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockSharesAuthorized_9d3f11ab-ffe6-47c9-bbdb-42d42bd019df_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Authorized share capital (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockSharesAuthorized_label_en-US" xlink:label="lab_us-gaap_CommonStockSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shares Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockSharesAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockSharesAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockSharesAuthorized" xlink:to="lab_us-gaap_CommonStockSharesAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueMeasurementsRecurringMember_3660f23b-ab9d-4e89-80c0-001c3b964f9e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasurementsRecurringMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Measurements, Recurring</link:label>
    <link:label id="lab_us-gaap_FairValueMeasurementsRecurringMember_label_en-US" xlink:label="lab_us-gaap_FairValueMeasurementsRecurringMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Recurring [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueMeasurementsRecurringMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueMeasurementsRecurringMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueMeasurementsRecurringMember" xlink:to="lab_us-gaap_FairValueMeasurementsRecurringMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_GainContingenciesTable_43b998a7-5f79-446a-8a15-15c623b00b36_terseLabel_en-US" xlink:label="lab_us-gaap_GainContingenciesTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gain Contingencies [Table]</link:label>
    <link:label id="lab_us-gaap_GainContingenciesTable_label_en-US" xlink:label="lab_us-gaap_GainContingenciesTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain Contingencies [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GainContingenciesTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GainContingenciesTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_GainContingenciesTable" xlink:to="lab_us-gaap_GainContingenciesTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset_e3a4bc10-1f82-4ede-a6d6-1fcda5333a5d_terseLabel_en-US" xlink:label="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Funds withheld receivable (Allowance for expected credit losses: 2025 - $9)</link:label>
    <link:label id="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset_fb079cc6-54f3-4a6e-a903-57f66bcd1cef_verboseLabel_en-US" xlink:label="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Funds withheld receivable</link:label>
    <link:label id="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset_label_en-US" xlink:label="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Funds Held under Reinsurance Agreements, Asset</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FundsHeldUnderReinsuranceAgreementsAsset"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:to="lab_us-gaap_FundsHeldUnderReinsuranceAgreementsAsset" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PremiumsReceivableAtCarryingValue_9b7a7e74-3fbe-4ab7-bb29-cb9c8a723a6f_verboseLabel_en-US" xlink:label="lab_us-gaap_PremiumsReceivableAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance balances receivable, net (includes $7,712 from related parties in 2025). Allowance for expected credit losses: 2025 - $229)</link:label>
    <link:label id="lab_us-gaap_PremiumsReceivableAtCarryingValue_952c0a8f-d4bc-478b-8b84-ac7f4c73f4d8_terseLabel_en-US" xlink:label="lab_us-gaap_PremiumsReceivableAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance balances receivable, net</link:label>
    <link:label id="lab_us-gaap_PremiumsReceivableAtCarryingValue_label_en-US" xlink:label="lab_us-gaap_PremiumsReceivableAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premiums Receivable, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PremiumsReceivableAtCarryingValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PremiumsReceivableAtCarryingValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PremiumsReceivableAtCarryingValue" xlink:to="lab_us-gaap_PremiumsReceivableAtCarryingValue" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_CoverAbstract_6082a525-582c-463b-9d7b-cb9a6b06014f_terseLabel_en-US" xlink:label="lab_dei_CoverAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cover [Abstract]</link:label>
    <link:label id="lab_dei_CoverAbstract_label_en-US" xlink:label="lab_dei_CoverAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cover [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_CoverAbstract" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_CoverAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_CoverAbstract" xlink:to="lab_dei_CoverAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember_571dec69-0db1-4ee2-b0a1-34490720b3b0_terseLabel_en-US" xlink:label="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Portion at Fair Value Measurement</link:label>
    <link:label id="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember_label_en-US" xlink:label="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Portion at Fair Value Measurement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PortionAtFairValueFairValueDisclosureMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:to="lab_us-gaap_PortionAtFairValueFairValueDisclosureMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock_bae43057-953c-4c3a-8de3-cd023b25c421_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Assets Held for Sale</link:label>
    <link:label id="lab_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Groups, Including Discontinued Operations, Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock" xlink:to="lab_us-gaap_DisposalGroupsIncludingDiscontinuedOperationsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_MaximumMember_8c58c7d2-40f2-499b-8d93-ef91ea3e2e28_terseLabel_en-US" xlink:label="lab_srt_MaximumMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Maximum</link:label>
    <link:label id="lab_srt_MaximumMember_label_en-US" xlink:label="lab_srt_MaximumMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Maximum [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_MaximumMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_MaximumMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_MaximumMember" xlink:to="lab_srt_MaximumMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount_e74d8b14-b9b2-49c3-af88-c6a3c98f0cb7_negatedTerseLabel_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Discount for present value</link:label>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount_label_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, Fair Value Discount</link:label>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount_documentation_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, Fair Value Discount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:href="kg-20250930.xsd#kg_LesseeOperatingLeaseLiabilityFairValueDiscount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:to="lab_kg_LesseeOperatingLeaseLiabilityFairValueDiscount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract_c2dcccdb-519e-4882-9b5a-c5dd42c32864_terseLabel_en-US" xlink:label="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loss and LAE</link:label>
    <link:label id="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract_label_en-US" xlink:label="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Policyholder Benefits and Claims Incurred, Net [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PolicyholderBenefitsAndClaimsIncurredNetAbstract"/>
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    <link:label id="lab_ecd_NonPeoNeoAvgTotalCompAmt_terseLabel_en-US" xlink:label="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Total Compensation Amount</link:label>
    <link:label id="lab_ecd_NonPeoNeoAvgTotalCompAmt_label_en-US" xlink:label="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Total Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonPeoNeoAvgTotalCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonPeoNeoAvgTotalCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonPeoNeoAvgTotalCompAmt" xlink:to="lab_ecd_NonPeoNeoAvgTotalCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentLineItems_e0834310-22a4-4a55-871f-1e7e15c7bed4_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument [Line Items]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentLineItems_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentLineItems" xlink:to="lab_us-gaap_DebtInstrumentLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionCancellationNoticePeriod_9629252d-69c0-46b5-a3ea-6c903181f829_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cancellation notice period</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionCancellationNoticePeriod_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Cancellation Notice Period</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionCancellationNoticePeriod_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Cancellation Notice Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionCancellationNoticePeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:to="lab_kg_RelatedPartyTransactionCancellationNoticePeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AdjToNonPeoNeoCompFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment to Non-PEO NEO Compensation Footnote</link:label>
    <link:label id="lab_ecd_AdjToNonPeoNeoCompFnTextBlock_label_en-US" xlink:label="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Adjustment to Non-PEO NEO Compensation Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AdjToNonPeoNeoCompFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:to="lab_ecd_AdjToNonPeoNeoCompFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DirectPremiumsEarned_4eb4edc9-8228-45c7-a301-625af0c069d9_terseLabel_en-US" xlink:label="lab_us-gaap_DirectPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Direct</link:label>
    <link:label id="lab_us-gaap_DirectPremiumsEarned_label_en-US" xlink:label="lab_us-gaap_DirectPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Direct Premiums Earned</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DirectPremiumsEarned" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DirectPremiumsEarned"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DirectPremiumsEarned" xlink:to="lab_us-gaap_DirectPremiumsEarned" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ExternalCreditRatingStandardPoorsMember_c5be8cb7-1ca0-4c33-8bcd-c48318393485_terseLabel_en-US" xlink:label="lab_srt_ExternalCreditRatingStandardPoorsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">External Credit Rating, Standard &amp; Poor's [Domain]</link:label>
    <link:label id="lab_srt_ExternalCreditRatingStandardPoorsMember_label_en-US" xlink:label="lab_srt_ExternalCreditRatingStandardPoorsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">External Credit Rating, Standard &amp; Poor's [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ExternalCreditRatingStandardPoorsMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ExternalCreditRatingStandardPoorsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ExternalCreditRatingStandardPoorsMember" xlink:to="lab_srt_ExternalCreditRatingStandardPoorsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Disqualification of Tax Benefits, Amount</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Disqualification of Tax Benefits, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:to="lab_ecd_ForgoneRecoveryDueToDisqualificationOfTaxBenefitsAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount_d79be6fb-876d-4dbd-b6c3-34a805674351_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Gain on bargain purchase</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount_718a1bb9-0f09-4cb4-9511-6b13350c1731_verboseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Gain on bargain purchase</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount_9a4f7467-e3df-4633-8d8b-d18aadba453f_negatedLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Bargain purchase gain</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Bargain Purchase, Gain Recognized, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:to="lab_us-gaap_BusinessCombinationBargainPurchaseGainRecognizedAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonRule10b51ArrTrmntdFlag_terseLabel_en-US" xlink:label="lab_ecd_NonRule10b51ArrTrmntdFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-Rule 10b5-1 Arrangement Terminated</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonRule10b51ArrTrmntdFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonRule10b51ArrTrmntdFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonRule10b51ArrTrmntdFlag" xlink:to="lab_ecd_NonRule10b51ArrTrmntdFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityShellCompany_52577a62-c2f6-4876-9a49-e3d230e2d850_terseLabel_en-US" xlink:label="lab_dei_EntityShellCompany" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Shell Company</link:label>
    <link:label id="lab_dei_EntityShellCompany_label_en-US" xlink:label="lab_dei_EntityShellCompany" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Shell Company</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityShellCompany" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityShellCompany"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityShellCompany" xlink:to="lab_dei_EntityShellCompany" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentTable_be3d527a-83e5-4dfd-afc8-9dc4f4114bb6_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Long-term Debt Instruments [Table]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentTable_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Long-Term Debt Instruments [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentTable" xlink:to="lab_us-gaap_DebtInstrumentTable" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_KestrelGroupMember_ea1579c7-4ed9-41d4-bcbd-155bfe0d6a76_terseLabel_en-US" xlink:label="lab_kg_KestrelGroupMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Kestrel Group</link:label>
    <link:label id="lab_kg_KestrelGroupMember_label_en-US" xlink:label="lab_kg_KestrelGroupMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Kestrel Group [Member]</link:label>
    <link:label id="lab_kg_KestrelGroupMember_documentation_en-US" xlink:label="lab_kg_KestrelGroupMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Kestrel Group</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_KestrelGroupMember" xlink:href="kg-20250930.xsd#kg_KestrelGroupMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_KestrelGroupMember" xlink:to="lab_kg_KestrelGroupMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_SummaryOfOtherInvestmentsTableTextBlock_18d52a89-8d2e-4adf-9c33-c6e761c3d5ca_terseLabel_en-US" xlink:label="lab_kg_SummaryOfOtherInvestmentsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Composition of Other Investments</link:label>
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    <link:label id="lab_kg_SummaryOfOtherInvestmentsTableTextBlock_documentation_en-US" xlink:label="lab_kg_SummaryOfOtherInvestmentsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Summary of Other Investments [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_SummaryOfOtherInvestmentsTableTextBlock" xlink:href="kg-20250930.xsd#kg_SummaryOfOtherInvestmentsTableTextBlock"/>
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    <link:label id="lab_us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve_5d7b4b4e-b6e9-411c-b953-04bd5a309ff1_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reserve for loss and loss adjustment expenses</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Liability for Claims and Claims Adjustment Expense Reserve</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInLiabilityForClaimsAndClaimsAdjustmentExpenseReserve"/>
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    <link:label id="lab_us-gaap_StatementOfStockholdersEquityAbstract_23e336af-a179-40f1-8d94-ebb4328117ad_terseLabel_en-US" xlink:label="lab_us-gaap_StatementOfStockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement of Stockholders' Equity [Abstract]</link:label>
    <link:label id="lab_us-gaap_StatementOfStockholdersEquityAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfStockholdersEquityAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Stockholders' Equity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfStockholdersEquityAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfStockholdersEquityAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfStockholdersEquityAbstract" xlink:to="lab_us-gaap_StatementOfStockholdersEquityAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingSegmentsMember_61c7ca96-b87f-438e-91e7-79062d437ee9_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingSegmentsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating Segments</link:label>
    <link:label id="lab_us-gaap_OperatingSegmentsMember_label_en-US" xlink:label="lab_us-gaap_OperatingSegmentsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Segments [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingSegmentsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingSegmentsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingSegmentsMember" xlink:to="lab_us-gaap_OperatingSegmentsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CommonStockShareholderVotingPowerPercentage_387a20ae-8b9e-48c7-825c-96400307a595_terseLabel_en-US" xlink:label="lab_kg_CommonStockShareholderVotingPowerPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Shareholder voting power, percentage</link:label>
    <link:label id="lab_kg_CommonStockShareholderVotingPowerPercentage_label_en-US" xlink:label="lab_kg_CommonStockShareholderVotingPowerPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shareholder Voting Power, Percentage</link:label>
    <link:label id="lab_kg_CommonStockShareholderVotingPowerPercentage_documentation_en-US" xlink:label="lab_kg_CommonStockShareholderVotingPowerPercentage" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Stock, Shareholder Voting Power, Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CommonStockShareholderVotingPowerPercentage" xlink:href="kg-20250930.xsd#kg_CommonStockShareholderVotingPowerPercentage"/>
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    <link:label id="lab_ecd_AwardTmgMnpiCnsdrdFlag_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgMnpiCnsdrdFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing MNPI Considered</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardTmgMnpiCnsdrdFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardTmgMnpiCnsdrdFlag"/>
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    <link:label id="lab_us-gaap_ReclassificationOutOfAccumulatedOtherComprehensiveIncomeTable_ce4b34ea-6822-44d6-aea7-b79289c6b6a9_terseLabel_en-US" xlink:label="lab_us-gaap_ReclassificationOutOfAccumulatedOtherComprehensiveIncomeTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reclassification out of Accumulated Other Comprehensive Income [Table]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReclassificationOutOfAccumulatedOtherComprehensiveIncomeTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReclassificationOutOfAccumulatedOtherComprehensiveIncomeTable"/>
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    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperations_1dfabc1e-8a4c-4b57-abf6-6c29bbc7025e_totalLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net (loss) income from continuing operations</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperations_label_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income (Loss) from Continuing Operations, Net of Tax, Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeLossFromContinuingOperations" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeLossFromContinuingOperations"/>
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    <link:label id="lab_kg_BusinessCombinationStockConversionRatio_6c001686-8aa1-42ee-a7aa-8773a1fd80a8_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationStockConversionRatio" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business combination, stock conversion ratio</link:label>
    <link:label id="lab_kg_BusinessCombinationStockConversionRatio_label_en-US" xlink:label="lab_kg_BusinessCombinationStockConversionRatio" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Stock Conversion Ratio</link:label>
    <link:label id="lab_kg_BusinessCombinationStockConversionRatio_documentation_en-US" xlink:label="lab_kg_BusinessCombinationStockConversionRatio" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Stock Conversion Ratio</link:label>
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    <link:label id="lab_kg_RelatedPartyTransactionPercentOfCedingCommissionPayablePeriodIncreaseDecrease_fdced4a0-8f85-4eaf-a8a3-625d72e79ca2_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfCedingCommissionPayablePeriodIncreaseDecrease" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percent of ceding commission payable, period increase</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfCedingCommissionPayablePeriodIncreaseDecrease_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfCedingCommissionPayablePeriodIncreaseDecrease" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent Of Ceding Commission Payable, Period Increase (Decrease)</link:label>
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    <link:label id="lab_us-gaap_ManagementFeeExpense_47b4b6ef-aaa1-42a5-8ab3-a636863a0ba3_terseLabel_en-US" xlink:label="lab_us-gaap_ManagementFeeExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expenses from related party transactions</link:label>
    <link:label id="lab_us-gaap_ManagementFeeExpense_label_en-US" xlink:label="lab_us-gaap_ManagementFeeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Management Fee Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ManagementFeeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ManagementFeeExpense"/>
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    <link:label id="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentBeforeTaxPortionAttributableToParent_6e043abb-b66c-422c-888e-57db35410858_terseLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentBeforeTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Foreign currency translation adjustment</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentBeforeTaxPortionAttributableToParent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeForeignCurrencyTransactionAndTranslationAdjustmentBeforeTaxPortionAttributableToParent"/>
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    <link:label id="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill_2f9fb2e7-3c7e-47a7-b6fe-a6258ae378c0_verboseLabel_en-US" xlink:label="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Intangible assets (includes $9,169 from related parties in 2025)</link:label>
    <link:label id="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill_4f610681-f72f-451b-9386-61c9b3f7b0c3_terseLabel_en-US" xlink:label="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible assets</link:label>
    <link:label id="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill_label_en-US" xlink:label="lab_us-gaap_IntangibleAssetsNetIncludingGoodwill" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Intangible Assets, Net (Including Goodwill)</link:label>
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    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInvestmentIncome_5e22b16a-62a8-46ae-9818-597f70242ae6_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedAccruedInvestmentIncome" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued investment income</link:label>
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    <link:label id="lab_ecd_AwardTmgMethodTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AwardTmgMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Timing Method</link:label>
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    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare_7a0a77d1-bca9-4619-ab1c-27605686d9a4_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basic (loss) income from continuing operations per share attributable to Kestrel common shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare_addc6097-63ff-42ab-aca2-597ad1917bca_verboseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Basic (loss) earnings from continuing operations per share attributable to Kestrel shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare_label_en-US" xlink:label="lab_us-gaap_IncomeLossFromContinuingOperationsPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income (Loss) from Continuing Operations, Per Basic Share</link:label>
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    <link:label id="lab_kg_AmtrustFinancialServicesIncMember_56eca3d0-7d6d-4c08-9938-8f696b99937f_terseLabel_en-US" xlink:label="lab_kg_AmtrustFinancialServicesIncMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">AmTrust Financial Services, Inc.</link:label>
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    <link:label id="lab_us-gaap_AwardTypeAxis_terseLabel_en-US" xlink:label="lab_us-gaap_AwardTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Award Type</link:label>
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    <link:label id="lab_us-gaap_AccumulatedTranslationAdjustmentMember_608d3a39-89f3-4e4b-a261-cecf1dd04069_terseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedTranslationAdjustmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Foreign currency translation</link:label>
    <link:label id="lab_us-gaap_AccumulatedTranslationAdjustmentMember_label_en-US" xlink:label="lab_us-gaap_AccumulatedTranslationAdjustmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accumulated Foreign Currency Adjustment Attributable to Parent [Member]</link:label>
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    <link:label id="lab_kg_SilverstoneVenture2Member_c0941b01-bca9-4fc3-8642-ddd646c0e298_terseLabel_en-US" xlink:label="lab_kg_SilverstoneVenture2Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Silverstone Venture 2</link:label>
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    <link:label id="lab_srt_EquityMethodInvesteeNameDomain_7fd7dac6-9399-440a-9b73-1005e605c119_terseLabel_en-US" xlink:label="lab_srt_EquityMethodInvesteeNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment, Name [Domain]</link:label>
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    <link:label id="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_8c0661e6-8217-4fb3-b908-1209fe2733f8_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Expense</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationOptionAndIncentivePlansPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Payment Arrangement [Policy Text Block]</link:label>
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    <link:label id="lab_dei_TradingSymbol_48793360-0301-4d6e-87b6-8bbf207fd041_terseLabel_en-US" xlink:label="lab_dei_TradingSymbol" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Symbol</link:label>
    <link:label id="lab_dei_TradingSymbol_label_en-US" xlink:label="lab_dei_TradingSymbol" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Symbol</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_TradingSymbol" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_TradingSymbol"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_TradingSymbol" xlink:to="lab_dei_TradingSymbol" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_996168a1-2c1a-43bd-bd85-a91c0ada85d2_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2029</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Four</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearFour" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationsPolicy_ba1f0dba-c911-462e-ab37-cc96f1d5b961_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationsPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accounting Treatment for the Combination</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationsPolicy_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationsPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationsPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationsPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationsPolicy" xlink:to="lab_us-gaap_BusinessCombinationsPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions_5199664f-f0c0-48b9-83e5-cf21d1598e53_terseLabel_en-US" xlink:label="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of securities in an unrealized loss position</link:label>
    <link:label id="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions_label_en-US" xlink:label="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Securities in Unrealized Loss Positions, Qualitative Disclosure, Number of Positions</link:label>
    <link:label id="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions_documentation_en-US" xlink:label="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Securities in Unrealized Loss Positions, Qualitative Disclosure, Number of Positions</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:href="kg-20250930.xsd#kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:to="lab_kg_FixedSecuritiesinUnrealizedLossPositionsQualitativeDisclosureNumberofPositions" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLossCarryforwardsTable_9f50bd33-ecf5-49f1-85a2-756d3613846c_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLossCarryforwardsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating Loss Carryforwards [Table]</link:label>
    <link:label id="lab_us-gaap_OperatingLossCarryforwardsTable_label_en-US" xlink:label="lab_us-gaap_OperatingLossCarryforwardsTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Loss Carryforwards [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLossCarryforwardsTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLossCarryforwardsTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLossCarryforwardsTable" xlink:to="lab_us-gaap_OperatingLossCarryforwardsTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross_867c2d24-748f-4c4c-88b5-2e311d2a48c4_terseLabel_en-US" xlink:label="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gross loss and LAE</link:label>
    <link:label id="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross_label_en-US" xlink:label="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Policyholder Benefits and Claims Incurred, Direct</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PolicyholderBenefitsAndClaimsIncurredGross"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross" xlink:to="lab_us-gaap_PolicyholderBenefitsAndClaimsIncurredGross" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities_8fcdb39d-336e-45cb-9e31-db4ef06ec3b1_terseLabel_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses and other liabilities (includes $68,508 and $0 from related parties in 2025 and 2024, respectively)</link:label>
    <link:label id="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities_59009262-a96c-4912-b14e-d827309985ee_verboseLabel_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses and other liabilities</link:label>
    <link:label id="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities_label_en-US" xlink:label="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accrued Liabilities and Other Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccruedLiabilitiesAndOtherLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:to="lab_us-gaap_AccruedLiabilitiesAndOtherLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue_ffdf3fe4-b795-4241-979d-21a6b9bdbbed_terseLabel_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair value of unrealized loss positions</link:label>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue_label_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investments, Continuous Unrealized Loss Position, Fair Value</link:label>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue_documentation_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investments, Continuous Unrealized Loss Position, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:href="kg-20250930.xsd#kg_InvestmentsContinuousUnrealizedLossPositionFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:to="lab_kg_InvestmentsContinuousUnrealizedLossPositionFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityAddressCityOrTown_0b142387-8b0a-4da3-8c29-1acde8a9c06c_terseLabel_en-US" xlink:label="lab_dei_EntityAddressCityOrTown" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, City or Town</link:label>
    <link:label id="lab_dei_EntityAddressCityOrTown_label_en-US" xlink:label="lab_dei_EntityAddressCityOrTown" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Address, City or Town</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressCityOrTown" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressCityOrTown"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityAddressCityOrTown" xlink:to="lab_dei_EntityAddressCityOrTown" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_0e00aca5-fd42-403d-8bf8-f3380c56b1a0_terseLabel_en-US" xlink:label="lab_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Effect of exchange rate changes on foreign currency cash, restricted cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents_label_en-US" xlink:label="lab_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Effect of Exchange Rate on Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:to="lab_us-gaap_EffectOfExchangeRateOnCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalents" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt_terseLabel_en-US" xlink:label="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Compensation Actually Paid Amount</link:label>
    <link:label id="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt_label_en-US" xlink:label="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Non-PEO NEO Average Compensation Actually Paid Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_NonPeoNeoAvgCompActuallyPaidAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:to="lab_ecd_NonPeoNeoAvgCompActuallyPaidAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Other Measure</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Other Measure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsOtherMeasureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsOtherMeasureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationConsiderationTransferred1_3e9e3ac4-4dba-44f8-8f80-29b9a2f33d53_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total consideration effectively transferred</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationConsiderationTransferred1_069e195a-204e-4656-8813-c2ce3e71f1fa_totalLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total consideration effectively transferred</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationConsiderationTransferred1_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Consideration Transferred</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationConsiderationTransferred1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:to="lab_us-gaap_BusinessCombinationConsiderationTransferred1" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityEmergingGrowthCompany_caa8869d-14ab-486d-a352-f1d80ab33a07_terseLabel_en-US" xlink:label="lab_dei_EntityEmergingGrowthCompany" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Emerging Growth Company</link:label>
    <link:label id="lab_dei_EntityEmergingGrowthCompany_label_en-US" xlink:label="lab_dei_EntityEmergingGrowthCompany" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Emerging Growth Company</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityEmergingGrowthCompany" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityEmergingGrowthCompany"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityEmergingGrowthCompany" xlink:to="lab_dei_EntityEmergingGrowthCompany" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract_7445a9c8-75e9-4ce1-b684-45b24ff817a7_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Changes in assets &#8211; (increase) decrease:</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Operating Assets [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInOperatingAssetsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract" xlink:to="lab_us-gaap_IncreaseDecreaseInOperatingAssetsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Change in Fair Value as of Vesting Date of Prior Year Equity Awards Vested in Covered Year</link:label>
    <link:label id="lab_ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember_label_en-US" xlink:label="lab_ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Change in Fair Value as of Vesting Date of Prior Year Equity Awards Vested in Covered Year [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ChngInFrValAsOfVstngDtOfPrrYrEqtyAwrdsVstdInCvrdYrMember"/>
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    <link:label id="lab_us-gaap_DeferredFinanceCostsNet_d6f790a5-184e-4869-bf36-b763fce8688d_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredFinanceCostsNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Less: unamortized fair value adjustment</link:label>
    <link:label id="lab_us-gaap_DeferredFinanceCostsNet_label_en-US" xlink:label="lab_us-gaap_DeferredFinanceCostsNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Issuance Costs, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredFinanceCostsNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredFinanceCostsNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredFinanceCostsNet" xlink:to="lab_us-gaap_DeferredFinanceCostsNet" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalents_8accb4cb-89bd-4408-ac2d-ed9c295de5be_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalents_3e05bfc1-3b34-45de-8909-a71c06d5a5f6_netLabel_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Restricted cash and cash equivalents, end of period</link:label>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalents_00792a78-7908-4c92-b4f0-a11bd0260d5b_verboseLabel_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Total restricted cash</link:label>
    <link:label id="lab_us-gaap_RestrictedCashAndCashEquivalents_label_en-US" xlink:label="lab_us-gaap_RestrictedCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash and Cash Equivalent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedCashAndCashEquivalents" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedCashAndCashEquivalents"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedCashAndCashEquivalents" xlink:to="lab_us-gaap_RestrictedCashAndCashEquivalents" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_DiversifiedReinsuranceSegmentMember_c3ab8fe2-852c-4a53-989a-4a046b5c9f47_terseLabel_en-US" xlink:label="lab_kg_DiversifiedReinsuranceSegmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Diversified Legacy Reinsurance</link:label>
    <link:label id="lab_kg_DiversifiedReinsuranceSegmentMember_757af6cd-94fc-4b89-9be3-848c286d85a6_verboseLabel_en-US" xlink:label="lab_kg_DiversifiedReinsuranceSegmentMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Diversified Reinsurance legacy business</link:label>
    <link:label id="lab_kg_DiversifiedReinsuranceSegmentMember_label_en-US" xlink:label="lab_kg_DiversifiedReinsuranceSegmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Diversified Reinsurance Segment [Member]</link:label>
    <link:label id="lab_kg_DiversifiedReinsuranceSegmentMember_documentation_en-US" xlink:label="lab_kg_DiversifiedReinsuranceSegmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Diversified Reinsurance Segment [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_DiversifiedReinsuranceSegmentMember" xlink:href="kg-20250930.xsd#kg_DiversifiedReinsuranceSegmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_DiversifiedReinsuranceSegmentMember" xlink:to="lab_kg_DiversifiedReinsuranceSegmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockSharesOutstanding_62247df6-9fe7-40a5-8618-a847e9153120_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares, outstanding (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockSharesOutstanding_label_en-US" xlink:label="lab_us-gaap_CommonStockSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shares, Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockSharesOutstanding" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockSharesOutstanding"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockSharesOutstanding" xlink:to="lab_us-gaap_CommonStockSharesOutstanding" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InsuranceAbstract_a5b225eb-e259-4224-95c8-2a008133aee4_terseLabel_en-US" xlink:label="lab_us-gaap_InsuranceAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Insurance [Abstract]</link:label>
    <link:label id="lab_us-gaap_InsuranceAbstract_label_en-US" xlink:label="lab_us-gaap_InsuranceAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insurance [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InsuranceAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InsuranceAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InsuranceAbstract" xlink:to="lab_us-gaap_InsuranceAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy_e68bce96-26e5-42b7-89b6-cf729aefe412_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance, amount retained, per policy</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance, Amount Retained, Per Policy</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy" xlink:to="lab_us-gaap_ReinsuranceRetentionAmountRetainedPerPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember_576f8d9d-8d72-4f30-8fe4-1bf72fac03e1_terseLabel_en-US" xlink:label="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investments transferred out related to Partial Termination Amendment and Commutation</link:label>
    <link:label id="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember_label_en-US" xlink:label="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AmTrust Quota Shares Reinsurance Agreement, Partial Termination Amendment [Member]</link:label>
    <link:label id="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember_documentation_en-US" xlink:label="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">AmTrust Quota Shares Reinsurance Agreement, Partial Termination Amendment [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:href="kg-20250930.xsd#kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:to="lab_kg_AmTrustQuotaSharesReinsuranceAgreementPartialTerminationAmendmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InternalRevenueServiceIRSMember_1df5f5ed-8807-4dc5-b472-afafec03ac81_terseLabel_en-US" xlink:label="lab_us-gaap_InternalRevenueServiceIRSMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Internal Revenue Service (IRS)</link:label>
    <link:label id="lab_us-gaap_InternalRevenueServiceIRSMember_label_en-US" xlink:label="lab_us-gaap_InternalRevenueServiceIRSMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Internal Revenue Service (IRS) [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InternalRevenueServiceIRSMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InternalRevenueServiceIRSMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InternalRevenueServiceIRSMember" xlink:to="lab_us-gaap_InternalRevenueServiceIRSMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Dividends or Other Earnings Paid on Equity Awards not Otherwise Reflected in Total Compensation for Covered Year</link:label>
    <link:label id="lab_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember_label_en-US" xlink:label="lab_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Dividends or Other Earnings Paid on Equity Awards not Otherwise Reflected in Total Compensation for Covered Year [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember" xlink:to="lab_ecd_DvddsOrOthrErngsPdOnEqtyAwrdsNtOthrwsRflctdInTtlCompForCvrdYrMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IntangibleAssetsFiniteLivedPolicy_c6aaa99c-fc5d-4b07-8bba-6ac53f545fb3_terseLabel_en-US" xlink:label="lab_us-gaap_IntangibleAssetsFiniteLivedPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible Assets</link:label>
    <link:label id="lab_us-gaap_IntangibleAssetsFiniteLivedPolicy_label_en-US" xlink:label="lab_us-gaap_IntangibleAssetsFiniteLivedPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Intangible Assets, Finite-Lived, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IntangibleAssetsFiniteLivedPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IntangibleAssetsFiniteLivedPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IntangibleAssetsFiniteLivedPolicy" xlink:to="lab_us-gaap_IntangibleAssetsFiniteLivedPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities_95474812-206b-497e-89c3-22224ae51627_terseLabel_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fixed maturities, available-for-sale, at fair value</link:label>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities_label_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group Including Discontinued Operation, Debt Securities</link:label>
    <link:label id="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities_documentation_en-US" xlink:label="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Disposal Group Including Discontinued Operation, Debt Securities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:href="kg-20250930.xsd#kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:to="lab_kg_DisposalGroupIncludingDiscontinuedOperationDebtSecurities" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued_39a86bc9-1dc2-49aa-9fd8-30a47971a329_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares issued or to be issued (in shares)</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued_label_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Contingent Consideration, Shares issued or To Be Issued</link:label>
    <link:label id="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued_documentation_en-US" xlink:label="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Contingent Consideration, Shares issued or To Be Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:href="kg-20250930.xsd#kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:to="lab_kg_BusinessCombinationContingentConsiderationSharesIssuedOrToBeIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateEffectivePercentage_2e1bf4f3-9eff-498a-af94-8592cbec090d_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateEffectivePercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Effective interest rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateEffectivePercentage_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateEffectivePercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Interest Rate, Effective Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentInterestRateEffectivePercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentInterestRateEffectivePercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentInterestRateEffectivePercentage" xlink:to="lab_us-gaap_DebtInstrumentInterestRateEffectivePercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityAddressPostalZipCode_dff66a46-f77d-44e8-af49-b9e5a4dc7276_terseLabel_en-US" xlink:label="lab_dei_EntityAddressPostalZipCode" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, Postal Zip Code</link:label>
    <link:label id="lab_dei_EntityAddressPostalZipCode_label_en-US" xlink:label="lab_dei_EntityAddressPostalZipCode" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Address, Postal Zip Code</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressPostalZipCode" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressPostalZipCode"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityAddressPostalZipCode" xlink:to="lab_dei_EntityAddressPostalZipCode" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_RestatementDeterminationDate_terseLabel_en-US" xlink:label="lab_ecd_RestatementDeterminationDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restatement Determination Date</link:label>
    <link:label id="lab_ecd_RestatementDeterminationDate_label_en-US" xlink:label="lab_ecd_RestatementDeterminationDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restatement Determination Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_RestatementDeterminationDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_RestatementDeterminationDate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_RestatementDeterminationDate" xlink:to="lab_ecd_RestatementDeterminationDate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeTaxExpenseBenefit_539773eb-e9af-4e8b-858e-f2793111479a_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxExpenseBenefit" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Less: income tax expense</link:label>
    <link:label id="lab_us-gaap_IncomeTaxExpenseBenefit_bd6822ea-e737-47ce-b15f-4e311a6a2ab9_negatedLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxExpenseBenefit" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Income tax expense</link:label>
    <link:label id="lab_us-gaap_IncomeTaxExpenseBenefit_039c73ca-c5e8-44be-85c6-bba148375f4a_verboseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxExpenseBenefit" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Income tax expense</link:label>
    <link:label id="lab_us-gaap_IncomeTaxExpenseBenefit_label_en-US" xlink:label="lab_us-gaap_IncomeTaxExpenseBenefit" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Tax Expense (Benefit)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeTaxExpenseBenefit" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeTaxExpenseBenefit"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeTaxExpenseBenefit" xlink:to="lab_us-gaap_IncomeTaxExpenseBenefit" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_b8edc55c-8f83-4ae9-a27a-079e4d24c88d_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fed funds rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Basis Spread on Variable Rate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentBasisSpreadOnVariableRate1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:to="lab_us-gaap_DebtInstrumentBasisSpreadOnVariableRate1" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare_087363ec-42ba-4ae9-943d-2413a5e9da23_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basic loss from discontinued operations per share attributable to Kestrel common shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare_97994e2e-b8e1-444c-829e-0cc21bb05938_verboseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Basic loss from discontinued operations per share attributable to Kestrel shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare_label_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income (Loss) from Discontinued Operations and Disposal of Discontinued Operations, Net of Tax, Per Basic Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerBasicShare"/>
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    <link:label id="lab_us-gaap_OtherAssets_d4a4eae4-9c54-4dbe-8fa8-a12784f536e0_terseLabel_en-US" xlink:label="lab_us-gaap_OtherAssets" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other assets</link:label>
    <link:label id="lab_us-gaap_OtherAssets_label_en-US" xlink:label="lab_us-gaap_OtherAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherAssets" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherAssets" xlink:to="lab_us-gaap_OtherAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue_e6a889ee-18d5-40fd-a3a7-a439e621d9da_periodStartLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Balance - beginning of period</link:label>
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    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue_label_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Asset Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetValue"/>
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    <link:label id="lab_srt_CreditRatingStandardPoorsAxis_7f59f5c6-59f2-4cfd-85b4-c03beaf674dd_terseLabel_en-US" xlink:label="lab_srt_CreditRatingStandardPoorsAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Credit Rating, Standard &amp; Poor's [Axis]</link:label>
    <link:label id="lab_srt_CreditRatingStandardPoorsAxis_label_en-US" xlink:label="lab_srt_CreditRatingStandardPoorsAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Credit Rating, Standard &amp; Poor's [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_CreditRatingStandardPoorsAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_CreditRatingStandardPoorsAxis"/>
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    <link:label id="lab_ecd_PnsnAdjsSvcCstMember_terseLabel_en-US" xlink:label="lab_ecd_PnsnAdjsSvcCstMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pension Adjustments Service Cost</link:label>
    <link:label id="lab_ecd_PnsnAdjsSvcCstMember_label_en-US" xlink:label="lab_ecd_PnsnAdjsSvcCstMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pension Adjustments Service Cost [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PnsnAdjsSvcCstMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PnsnAdjsSvcCstMember"/>
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    <link:label id="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis_24c4dbaf-0e9f-4a87-b13a-ba2c2f727707_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Hierarchy and NAV [Axis]</link:label>
    <link:label id="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis_label_en-US" xlink:label="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Hierarchy and NAV [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueByFairValueHierarchyLevelAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:to="lab_us-gaap_FairValueByFairValueHierarchyLevelAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DiscontinuedOperationsHeldforsaleMember_e15c3e1d-de29-43a5-80ca-3e7413dd2220_terseLabel_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsHeldforsaleMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discontinued Operations, Held-for-sale</link:label>
    <link:label id="lab_us-gaap_DiscontinuedOperationsHeldforsaleMember_label_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsHeldforsaleMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Discontinued Operations, Held-for-Sale [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DiscontinuedOperationsHeldforsaleMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DiscontinuedOperationsHeldforsaleMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DiscontinuedOperationsHeldforsaleMember" xlink:to="lab_us-gaap_DiscontinuedOperationsHeldforsaleMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract_c83fedc4-ca57-4229-8153-4a85c3fae9b6_terseLabel_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discontinued Operations and Disposal Groups [Abstract]</link:label>
    <link:label id="lab_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract_label_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Discontinued Operations and Disposal Groups [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract" xlink:to="lab_us-gaap_DiscontinuedOperationsAndDisposalGroupsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentManagementFeesMember_296b9b69-af02-408b-8ace-fac0d62f1777_terseLabel_en-US" xlink:label="lab_kg_InvestmentManagementFeesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Management Fee</link:label>
    <link:label id="lab_kg_InvestmentManagementFeesMember_label_en-US" xlink:label="lab_kg_InvestmentManagementFeesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Management Fees [Member]</link:label>
    <link:label id="lab_kg_InvestmentManagementFeesMember_documentation_en-US" xlink:label="lab_kg_InvestmentManagementFeesMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investment Management Fees [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentManagementFeesMember" xlink:href="kg-20250930.xsd#kg_InvestmentManagementFeesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentManagementFeesMember" xlink:to="lab_kg_InvestmentManagementFeesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockholdersEquityPolicyTextBlock_5dc2e648-6009-4fde-a42c-6564945893bd_terseLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquityPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Treasury Shares</link:label>
    <link:label id="lab_us-gaap_StockholdersEquityPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_StockholdersEquityPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stockholders' Equity, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquityPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquityPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockholdersEquityPolicyTextBlock" xlink:to="lab_us-gaap_StockholdersEquityPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock_terseLabel_en-US" xlink:label="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Company Selected Measure</link:label>
    <link:label id="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock_label_en-US" xlink:label="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Compensation Actually Paid vs. Company Selected Measure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:to="lab_ecd_CompActuallyPaidVsCoSelectedMeasureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionSolvencyRatio_1f985f75-3ce9-4d7c-b191-26225822acc1_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionSolvencyRatio" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Solvency ratio</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionSolvencyRatio_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionSolvencyRatio" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Solvency Ratio</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionSolvencyRatio_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionSolvencyRatio" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Solvency Ratio</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionSolvencyRatio" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionSolvencyRatio"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionSolvencyRatio" xlink:to="lab_kg_RelatedPartyTransactionSolvencyRatio" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllTradingArrangementsMember_terseLabel_en-US" xlink:label="lab_ecd_AllTradingArrangementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Trading Arrangements</link:label>
    <link:label id="lab_ecd_AllTradingArrangementsMember_label_en-US" xlink:label="lab_ecd_AllTradingArrangementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Trading Arrangements [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllTradingArrangementsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllTradingArrangementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllTradingArrangementsMember" xlink:to="lab_ecd_AllTradingArrangementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_EqtyAwrdsAdjFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments, Footnote</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_EqtyAwrdsAdjFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_EqtyAwrdsAdjFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_EqtyAwrdsAdjFnTextBlock" xlink:to="lab_ecd_EqtyAwrdsAdjFnTextBlock" xlink:type="arc" order="1"/>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TotalShareholderRtnVsPeerGroupTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:to="lab_ecd_TotalShareholderRtnVsPeerGroupTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentRedemptionPricePercentage_36bbd1db-e926-4593-a09c-34a803bd94cb_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentRedemptionPricePercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Redemption price, percentage</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentRedemptionPricePercentage_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentRedemptionPricePercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Redemption Price, Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentRedemptionPricePercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentRedemptionPricePercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentRedemptionPricePercentage" xlink:to="lab_us-gaap_DebtInstrumentRedemptionPricePercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty_d84fea95-8315-4309-9f78-8c81495eff5a_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collateral held by related party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Collateral Held by Related Party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Collateral Held by Related Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionCollateralHeldByRelatedParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:to="lab_kg_RelatedPartyTransactionCollateralHeldByRelatedParty" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_NumberOfDirectorsSelectedByAffiliate_4141774d-bf8a-4df4-b555-52c9336db801_terseLabel_en-US" xlink:label="lab_kg_NumberOfDirectorsSelectedByAffiliate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of directors, selected by affiliate</link:label>
    <link:label id="lab_kg_NumberOfDirectorsSelectedByAffiliate_label_en-US" xlink:label="lab_kg_NumberOfDirectorsSelectedByAffiliate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number of Directors, Selected by Affiliate</link:label>
    <link:label id="lab_kg_NumberOfDirectorsSelectedByAffiliate_documentation_en-US" xlink:label="lab_kg_NumberOfDirectorsSelectedByAffiliate" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number of Directors, Selected by Affiliate</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_NumberOfDirectorsSelectedByAffiliate" xlink:href="kg-20250930.xsd#kg_NumberOfDirectorsSelectedByAffiliate"/>
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    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt_643ed9d2-3243-4624-8b15-ae77cfe7605e_negatedLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Senior notes - at fair value</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt_label_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Long-Term Debt</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt_documentation_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Long-Term Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLongTermDebt"/>
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    <link:label id="lab_kg_RelatedPartyTransactionPercentageOfCollateralOnExposure_d86a7250-1db1-4d5f-bc23-06c17c3fd7f3_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentageOfCollateralOnExposure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collateral on exposure</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommitmentsAndContingenciesDisclosureAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommitmentsAndContingenciesDisclosureAbstract"/>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivities_9f53b0ef-c8d8-443e-8019-e846522f8235_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash used in operating activities</link:label>
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    <link:label id="lab_us-gaap_ProceedsFromSaleOfAvailableForSaleSecuritiesDebt_b2ebaae4-3412-4709-8739-566a1e02de57_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromSaleOfAvailableForSaleSecuritiesDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from sales of fixed maturities&#160;</link:label>
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    <link:label id="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRecovery_fdc7dde6-002c-44e8-a779-2fda9a1aa28e_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRecovery" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Decrease in allowance for expected credit losses on reinsurance recoverable where credit losses were previously recognized</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRecovery" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRecovery"/>
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    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales_f379d627-fc18-4806-aa4c-060030801e33_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Sales</link:label>
    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales_label_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Measurement with Unobservable Inputs Reconciliation, Recurring Basis, Asset, Sales</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales"/>
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    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsOther_d6f56349-1626-4436-9a7f-5b26e931101b_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCurrentAssetsOther" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other assets</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfEquityMethodInvestmentsLineItems_5e5d4aba-4660-4388-8c1b-bd82cad4b24c_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfEquityMethodInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Equity Method Investments [Line Items]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfEquityMethodInvestmentsLineItems_label_en-US" xlink:label="lab_us-gaap_ScheduleOfEquityMethodInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Equity Method Investments [Line Items]</link:label>
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    <link:label id="lab_ecd_PeoTotalCompAmt_label_en-US" xlink:label="lab_ecd_PeoTotalCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">PEO Total Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoTotalCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoTotalCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeoTotalCompAmt" xlink:to="lab_ecd_PeoTotalCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongtermDebtTypeAxis_17a2cc54-b3bf-4038-b5b1-f44de408466e_terseLabel_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-term Debt, Type [Axis]</link:label>
    <link:label id="lab_us-gaap_LongtermDebtTypeAxis_label_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Type [Axis]</link:label>
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    <link:label id="lab_us-gaap_DerivativeLiabilityMeasurementInput_2f70b84b-57cb-42b6-80f5-6e36df178665_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeLiabilityMeasurementInput" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discount rates</link:label>
    <link:label id="lab_us-gaap_DerivativeLiabilityMeasurementInput_label_en-US" xlink:label="lab_us-gaap_DerivativeLiabilityMeasurementInput" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Derivative Liability, Measurement Input</link:label>
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    <link:label id="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract_eb147330-d2f1-4a02-a379-e957da12706f_terseLabel_en-US" xlink:label="lab_us-gaap_WeightedAverageNumberOfSharesOutstandingBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Denominator:</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfInvestmentsAbstract_2e4fb8ce-d31e-4539-8f97-6033ebe2f814_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfInvestmentsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Investments [Abstract]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfInvestmentsAbstract_label_en-US" xlink:label="lab_us-gaap_ScheduleOfInvestmentsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Investments [Abstract]</link:label>
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    <link:label id="lab_us-gaap_StatementEquityComponentsAxis_ae24605f-808d-4ab4-bb49-546d72cdc4bf_terseLabel_en-US" xlink:label="lab_us-gaap_StatementEquityComponentsAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Components [Axis]</link:label>
    <link:label id="lab_us-gaap_StatementEquityComponentsAxis_label_en-US" xlink:label="lab_us-gaap_StatementEquityComponentsAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Components [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementEquityComponentsAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementEquityComponentsAxis"/>
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    <link:label id="lab_kg_OCIDebtSecuritiesHeldForSaleUnrealizedHoldingGainLossBeforeAdjustmentAndTax_dfdb68e0-7a47-4201-9908-7a0c1a1e75e8_terseLabel_en-US" xlink:label="lab_kg_OCIDebtSecuritiesHeldForSaleUnrealizedHoldingGainLossBeforeAdjustmentAndTax" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net unrealized gains on held for sale fixed maturities</link:label>
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    <link:label id="lab_us-gaap_NumberOfReportableSegments_d5fff4b0-7d21-4345-b5b6-540d5f6580ad_terseLabel_en-US" xlink:label="lab_us-gaap_NumberOfReportableSegments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of reporting segments</link:label>
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    <link:label id="lab_dei_EntityAddressCountry_502ee8b2-427f-4e3e-b151-3eaeffa5f228_terseLabel_en-US" xlink:label="lab_dei_EntityAddressCountry" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, Country</link:label>
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    <link:label id="lab_us-gaap_NetIncomeLossFromDiscontinuedOperationsAvailableToCommonShareholdersBasic_bee09f74-cfa1-4db9-95c7-4ffc2234dc7e_terseLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossFromDiscontinuedOperationsAvailableToCommonShareholdersBasic" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amount of loss from discontinued operations allocated to participating common shareholders, basic</link:label>
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    <link:label id="lab_ecd_IndividualAxis_terseLabel_en-US" xlink:label="lab_ecd_IndividualAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Individual:</link:label>
    <link:label id="lab_ecd_IndividualAxis_label_en-US" xlink:label="lab_ecd_IndividualAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Individual [Axis]</link:label>
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    <link:label id="lab_us-gaap_ReinsuranceRetentionPolicyTable_87b9830d-8f58-45cd-afe2-ac234e0e1393_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRetentionPolicyTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance Retention Policy [Table]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRetentionPolicyTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRetentionPolicyTable"/>
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    <link:label id="lab_us-gaap_IncomeTaxAuthorityDomain_dd2de3b3-02dd-4400-ad3f-495b9d126438_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxAuthorityDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Income Tax Authority [Domain]</link:label>
    <link:label id="lab_us-gaap_IncomeTaxAuthorityDomain_label_en-US" xlink:label="lab_us-gaap_IncomeTaxAuthorityDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Tax Jurisdiction [Domain]</link:label>
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    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments_d2a66c93-a711-4118-bf96-db7431ea5e2a_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity method investments</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments" xlink:to="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedEquityMethodInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PaymentsOfDividendsCommonStock_54d6f099-7eae-46cb-b254-5db7675b8a12_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_PaymentsOfDividendsCommonStock" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Dividends paid &#8211; Kestrel equityholders</link:label>
    <link:label id="lab_us-gaap_PaymentsOfDividendsCommonStock_label_en-US" xlink:label="lab_us-gaap_PaymentsOfDividendsCommonStock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments of Ordinary Dividends, Common Stock</link:label>
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    <link:label id="lab_kg_BusinessCombinationAcquisitionOptionTerm_50e9eccd-79a6-417d-a619-c0374764b57e_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Acquisition option term</link:label>
    <link:label id="lab_kg_BusinessCombinationAcquisitionOptionTerm_label_en-US" xlink:label="lab_kg_BusinessCombinationAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Acquisition Option Term</link:label>
    <link:label id="lab_kg_BusinessCombinationAcquisitionOptionTerm_documentation_en-US" xlink:label="lab_kg_BusinessCombinationAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Acquisition Option Term</link:label>
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    <link:label id="lab_kg_OtherInvestmentMember_c584f4ce-4b71-4623-91de-b9642ff761ff_terseLabel_en-US" xlink:label="lab_kg_OtherInvestmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other investments</link:label>
    <link:label id="lab_kg_OtherInvestmentMember_label_en-US" xlink:label="lab_kg_OtherInvestmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Investment [Member]</link:label>
    <link:label id="lab_kg_OtherInvestmentMember_documentation_en-US" xlink:label="lab_kg_OtherInvestmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Other Investment</link:label>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateAmortizedCost_4883de67-e95d-49ef-8315-206eeb9531c0_verboseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateAmortizedCost" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Amortized cost of securities without single maturities</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateAmortizedCost_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateAmortizedCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Maturity, without Single Maturity Date, Amortized Cost</link:label>
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    <link:label id="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryExplanationOfImpracticabilityTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery, Explanation of Impracticability</link:label>
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    <link:label id="lab_us-gaap_MinorityInterestOwnershipPercentageByNoncontrollingOwners_43f8e1a4-a33d-4407-b64d-eeca8ef12ac1_terseLabel_en-US" xlink:label="lab_us-gaap_MinorityInterestOwnershipPercentageByNoncontrollingOwners" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Minority interest</link:label>
    <link:label id="lab_us-gaap_MinorityInterestOwnershipPercentageByNoncontrollingOwners_label_en-US" xlink:label="lab_us-gaap_MinorityInterestOwnershipPercentageByNoncontrollingOwners" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Subsidiary, Ownership Percentage, Noncontrolling Owner</link:label>
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    <link:label id="lab_us-gaap_DeferredTaxAssetsDeferredIncome_fbcbf34d-0ff6-47d0-9ee3-61b5549c06af_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredTaxAssetsDeferredIncome" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Deferred income tax</link:label>
    <link:label id="lab_us-gaap_DeferredTaxAssetsDeferredIncome_label_en-US" xlink:label="lab_us-gaap_DeferredTaxAssetsDeferredIncome" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deferred Tax Assets, Deferred Income</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredTaxAssetsDeferredIncome" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredTaxAssetsDeferredIncome"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredTaxAssetsDeferredIncome" xlink:to="lab_us-gaap_DeferredTaxAssetsDeferredIncome" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty_7e23aaf3-6312-430b-9785-99a6bd1f1287_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net loan receivable from related party</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty_label_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Net Loan Receivable From Related Party</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty_documentation_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed Net Loan Receivable From Related Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:to="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNetLoanReceivableFromRelatedParty" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale_fa8966b6-c9af-4f3e-840d-d3eb713919d5_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fixed maturities, available-for-sale, at fair value</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale_label_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Fixed Maturities, Available For Sale</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale_documentation_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Fixed Maturities, Available For Sale</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:to="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedFixedMaturitiesAvailableForSale" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeAxis_ca2ba416-892e-49c5-b5b8-a7b70eb1c527_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Axis]</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeAxis_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Axis]</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeAxis_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeAxis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionByTypeAxis" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionByTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionByTypeAxis" xlink:to="lab_kg_RelatedPartyTransactionByTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_OwnershipDomain_1e12780c-b905-4ccc-9c93-36e8095d30a6_terseLabel_en-US" xlink:label="lab_srt_OwnershipDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Ownership [Domain]</link:label>
    <link:label id="lab_srt_OwnershipDomain_label_en-US" xlink:label="lab_srt_OwnershipDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Ownership [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_OwnershipDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_OwnershipDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_OwnershipDomain" xlink:to="lab_srt_OwnershipDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService_71b9692b-66e7-4451-b3ed-fa183b7f9acd_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Portion of the Maiden awards attributable to pre-combination service</link:label>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService_label_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination Consideration Transferred Attributable to Pre Combination Service</link:label>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService_documentation_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination Consideration Transferred Attributable to Pre Combination Service</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:href="kg-20250930.xsd#kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:to="lab_kg_BusinessCombinationConsiderationTransferredAttributableToPreCombinationService" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InterestPayableCurrent_74443b4b-18f7-451b-ba05-83eff22f04e2_terseLabel_en-US" xlink:label="lab_us-gaap_InterestPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued interest</link:label>
    <link:label id="lab_us-gaap_InterestPayableCurrent_label_en-US" xlink:label="lab_us-gaap_InterestPayableCurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Payable, Current</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestPayableCurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestPayableCurrent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestPayableCurrent" xlink:to="lab_us-gaap_InterestPayableCurrent" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityCentralIndexKey_d7ea98ea-d957-4cac-8786-bddf4a3638ee_terseLabel_en-US" xlink:label="lab_dei_EntityCentralIndexKey" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Central Index Key</link:label>
    <link:label id="lab_dei_EntityCentralIndexKey_label_en-US" xlink:label="lab_dei_EntityCentralIndexKey" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Central Index Key</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityCentralIndexKey" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityCentralIndexKey"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityCentralIndexKey" xlink:to="lab_dei_EntityCentralIndexKey" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeoName_terseLabel_en-US" xlink:label="lab_ecd_PeoName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO Name</link:label>
    <link:label id="lab_ecd_PeoName_label_en-US" xlink:label="lab_ecd_PeoName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">PEO Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeoName" xlink:to="lab_ecd_PeoName" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_Revenues_d6271935-e4d8-48bb-a5ce-e33eb1e7cd92_totalLabel_en-US" xlink:label="lab_us-gaap_Revenues" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total revenues</link:label>
    <link:label id="lab_us-gaap_Revenues_label_en-US" xlink:label="lab_us-gaap_Revenues" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenues</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Revenues" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Revenues"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_Revenues" xlink:to="lab_us-gaap_Revenues" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prior Year End Fair Value of Equity Awards Granted in Any Prior Year that Fail to Meet Applicable Vesting Conditions During Covered Year</link:label>
    <link:label id="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember_label_en-US" xlink:label="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Prior Year End Fair Value of Equity Awards Granted in Any Prior Year that Fail to Meet Applicable Vesting Conditions During Covered Year [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:to="lab_ecd_FrValAsOfPrrYrEndOfEqtyAwrdsGrntdInPrrYrsFldVstngCondsDrngCvrdYrMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_OutstandingAggtErrCompAmt_terseLabel_en-US" xlink:label="lab_ecd_OutstandingAggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Outstanding Aggregate Erroneous Compensation Amount</link:label>
    <link:label id="lab_ecd_OutstandingAggtErrCompAmt_label_en-US" xlink:label="lab_ecd_OutstandingAggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Aggregate Erroneous Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingAggtErrCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingAggtErrCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OutstandingAggtErrCompAmt" xlink:to="lab_ecd_OutstandingAggtErrCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RealizedAndUnrealizedGainOnInvestments_b5da7dcf-f77b-42d3-ad28-9e9d9a8c74b5_totalLabel_en-US" xlink:label="lab_kg_RealizedAndUnrealizedGainOnInvestments" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net realized and unrealized gains (losses) on investment, gross gains</link:label>
    <link:label id="lab_kg_RealizedAndUnrealizedGainOnInvestments_label_en-US" xlink:label="lab_kg_RealizedAndUnrealizedGainOnInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Realized And Unrealized Gain On Investments</link:label>
    <link:label id="lab_kg_RealizedAndUnrealizedGainOnInvestments_documentation_en-US" xlink:label="lab_kg_RealizedAndUnrealizedGainOnInvestments" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Realized And Unrealized Gain On Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RealizedAndUnrealizedGainOnInvestments" xlink:href="kg-20250930.xsd#kg_RealizedAndUnrealizedGainOnInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RealizedAndUnrealizedGainOnInvestments" xlink:to="lab_kg_RealizedAndUnrealizedGainOnInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_9e6746b9-2872-4f83-a830-5fcd19cb32bc_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Segment Reporting Information, by Segment [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Segment Reporting Information, by Segment [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:to="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTable" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_QuotaShareReinsuranceAgreementsMember_3e314140-d273-4aec-8196-4aafc9239f0e_terseLabel_en-US" xlink:label="lab_kg_QuotaShareReinsuranceAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Quota Share Reinsurance Agreements</link:label>
    <link:label id="lab_kg_QuotaShareReinsuranceAgreementsMember_label_en-US" xlink:label="lab_kg_QuotaShareReinsuranceAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Quota Share Reinsurance Agreements [Member]</link:label>
    <link:label id="lab_kg_QuotaShareReinsuranceAgreementsMember_documentation_en-US" xlink:label="lab_kg_QuotaShareReinsuranceAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Quota Share Reinsurance Agreements [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_QuotaShareReinsuranceAgreementsMember" xlink:href="kg-20250930.xsd#kg_QuotaShareReinsuranceAgreementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_QuotaShareReinsuranceAgreementsMember" xlink:to="lab_kg_QuotaShareReinsuranceAgreementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PremiumsWrittenNet_0541c79d-a049-4eae-8d37-4a8c12624718_terseLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net premiums written</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenNet_180b25dd-87ee-4531-a059-353797011641_verboseLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenNet" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Net premiums written</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenNet_e26a338a-3cf0-46dc-9046-f461f7f5e4cb_netLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenNet" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Net premiums written</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenNet_5553195f-7e25-4b96-ba71-064c27073761_totalLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenNet" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenNet_label_en-US" xlink:label="lab_us-gaap_PremiumsWrittenNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premiums Written, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PremiumsWrittenNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PremiumsWrittenNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PremiumsWrittenNet" xlink:to="lab_us-gaap_PremiumsWrittenNet" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts_e30e7839-a5da-45bf-8c5f-304179991a21_terseLabel_en-US" xlink:label="lab_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance balances receivable, net, allowance for expected credit losses</link:label>
    <link:label id="lab_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts_label_en-US" xlink:label="lab_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premium Receivable, Allowance for Credit Loss</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts" xlink:to="lab_us-gaap_PremiumsReceivableAllowanceForDoubtfulAccounts" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AwardExrcPrice_terseLabel_en-US" xlink:label="lab_ecd_AwardExrcPrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Exercise Price</link:label>
    <link:label id="lab_ecd_AwardExrcPrice_label_en-US" xlink:label="lab_ecd_AwardExrcPrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Exercise Price</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardExrcPrice" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardExrcPrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AwardExrcPrice" xlink:to="lab_ecd_AwardExrcPrice" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityFilerCategory_7b83032c-5e63-45f7-92cf-7b6b06564a24_terseLabel_en-US" xlink:label="lab_dei_EntityFilerCategory" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Filer Category</link:label>
    <link:label id="lab_dei_EntityFilerCategory_label_en-US" xlink:label="lab_dei_EntityFilerCategory" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Filer Category</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityFilerCategory" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityFilerCategory"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityFilerCategory" xlink:to="lab_dei_EntityFilerCategory" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale_bd476f8b-49dd-4f92-aafb-7008886a7a34_terseLabel_en-US" xlink:label="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Less: change in cash and cash equivalents held for sale</link:label>
    <link:label id="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale_label_en-US" xlink:label="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents, Held-for-Sale</link:label>
    <link:label id="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale_documentation_en-US" xlink:label="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Cash, Cash Equivalents, Restricted Cash, and Restricted Cash Equivalents, Held-for-Sale</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:href="kg-20250930.xsd#kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:to="lab_kg_IncreaseDecreaseInCashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsHeldForSale" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_DirectLendingEntitiesMember_c5927f4b-6f4e-4796-a4f6-501f46fe46e7_terseLabel_en-US" xlink:label="lab_kg_DirectLendingEntitiesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investments in direct lending entities</link:label>
    <link:label id="lab_kg_DirectLendingEntitiesMember_7d1d134a-dfc9-4359-b8d7-67fec1082fff_verboseLabel_en-US" xlink:label="lab_kg_DirectLendingEntitiesMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Other investments - Investments in direct lending entities</link:label>
    <link:label id="lab_kg_DirectLendingEntitiesMember_label_en-US" xlink:label="lab_kg_DirectLendingEntitiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Direct Lending Entities [Member]</link:label>
    <link:label id="lab_kg_DirectLendingEntitiesMember_documentation_en-US" xlink:label="lab_kg_DirectLendingEntitiesMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Direct Lending Entities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_DirectLendingEntitiesMember" xlink:href="kg-20250930.xsd#kg_DirectLendingEntitiesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_DirectLendingEntitiesMember" xlink:to="lab_kg_DirectLendingEntitiesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents_23898293-3956-4e53-a093-28917586a0e2_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents_label_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Cash and Cash Equivalents</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents" xlink:to="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationCashAndCashEquivalents" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember_terseLabel_en-US" xlink:label="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Vesting Date Fair Value of Equity Awards Granted and Vested in Covered Year</link:label>
    <link:label id="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember_label_en-US" xlink:label="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Vesting Date Fair Value of Equity Awards Granted and Vested in Covered Year [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:to="lab_ecd_VstngDtFrValOfEqtyAwrdsGrntdAndVstdInCvrdYrMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PremiumsWrittenGross_82e9a280-8660-40c3-89e2-fee98d4fdfd3_terseLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gross premiums written</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenGross_091f5d7e-a4bc-43e2-b54f-44a2619c7a81_verboseLabel_en-US" xlink:label="lab_us-gaap_PremiumsWrittenGross" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Gross premiums written</link:label>
    <link:label id="lab_us-gaap_PremiumsWrittenGross_label_en-US" xlink:label="lab_us-gaap_PremiumsWrittenGross" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premiums Written, Gross</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PremiumsWrittenGross" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PremiumsWrittenGross"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PremiumsWrittenGross" xlink:to="lab_us-gaap_PremiumsWrittenGross" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityAddressAddressLine1_ccd7fd2f-3a89-469c-a64c-6be618af9816_terseLabel_en-US" xlink:label="lab_dei_EntityAddressAddressLine1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Address, Address Line One</link:label>
    <link:label id="lab_dei_EntityAddressAddressLine1_label_en-US" xlink:label="lab_dei_EntityAddressAddressLine1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Address, Address Line One</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityAddressAddressLine1" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityAddressAddressLine1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityAddressAddressLine1" xlink:to="lab_dei_EntityAddressAddressLine1" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EquityMethodInvestmentUnrealizedGainLoss_9be88f83-df08-43e8-8fa4-3aa8a5c7251f_terseLabel_en-US" xlink:label="lab_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net unrealized losses on held for sale AFS investments</link:label>
    <link:label id="lab_kg_EquityMethodInvestmentUnrealizedGainLoss_label_en-US" xlink:label="lab_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investment, Unrealized Gain (Loss)</link:label>
    <link:label id="lab_kg_EquityMethodInvestmentUnrealizedGainLoss_documentation_en-US" xlink:label="lab_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Equity Method Investment, Unrealized Gain (Loss)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:href="kg-20250930.xsd#kg_EquityMethodInvestmentUnrealizedGainLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:to="lab_kg_EquityMethodInvestmentUnrealizedGainLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember_cc772102-4583-410c-bd37-94247461ef22_terseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated other comprehensive loss</link:label>
    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember_aa582a6e-b61e-4915-ba60-22ef0022ad16_verboseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Total</link:label>
    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember_label_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AOCI Attributable to Parent [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccumulatedOtherComprehensiveIncomeMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:to="lab_us-gaap_AccumulatedOtherComprehensiveIncomeMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities_a1ac1ad4-ee39-4f98-93d4-8337ba1f0884_verboseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses and other liabilities</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Accrued Liabilities and Other Operating Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities" xlink:to="lab_us-gaap_IncreaseDecreaseInAccruedLiabilitiesAndOtherOperatingLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InvestmentTypeCategorizationMember_26b1197f-0fd9-4f73-8afe-e7e571e19bde_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentTypeCategorizationMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investments [Domain]</link:label>
    <link:label id="lab_us-gaap_InvestmentTypeCategorizationMember_label_en-US" xlink:label="lab_us-gaap_InvestmentTypeCategorizationMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investments [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentTypeCategorizationMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentTypeCategorizationMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentTypeCategorizationMember" xlink:to="lab_us-gaap_InvestmentTypeCategorizationMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments_acd34901-9183-4a54-a2b1-06fcf9461314_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverable on unpaid losses (Allowance for expected credit losses: 2025 - $1,459)</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments_8a6f0d4a-a24b-46c4-acd3-b79406a9a817_verboseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverable on unpaid losses</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable for Unpaid Claims and Claims Adjustments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:to="lab_us-gaap_ReinsuranceRecoverableForUnpaidClaimsAndClaimsAdjustments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract_0ec9b4ae-70f4-408b-96e6-20f93d99c1bf_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination, Asset Acquisition, Transaction between Entities under Common Control, and Joint Venture Formation [Abstract]</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Asset Acquisition, Transaction between Entities under Common Control, and Joint Venture Formation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationAndAssetAcquisitionAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract" xlink:to="lab_us-gaap_BusinessCombinationAndAssetAcquisitionAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_OwnershipAxis_f3032f45-7115-41cf-9b8c-00a30b8b3678_terseLabel_en-US" xlink:label="lab_srt_OwnershipAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Ownership [Axis]</link:label>
    <link:label id="lab_srt_OwnershipAxis_label_en-US" xlink:label="lab_srt_OwnershipAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Ownership [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_OwnershipAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_OwnershipAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_OwnershipAxis" xlink:to="lab_srt_OwnershipAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments_4d6d7ed7-19ad-4ab9-b8b0-5de5c8b61693_terseLabel_en-US" xlink:label="lab_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from sale and redemption of other investments</link:label>
    <link:label id="lab_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments_label_en-US" xlink:label="lab_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Proceeds from Sale and Maturity of Other Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments" xlink:to="lab_us-gaap_ProceedsFromSaleAndMaturityOfOtherInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems_c4f0205a-24b7-496d-961e-2a5031d5fe8b_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items]</link:label>
    <link:label id="lab_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems_label_en-US" xlink:label="lab_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Statement, Balance Sheet and Additional Disclosures by Disposal Groups, Including Discontinued Operations [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems" xlink:to="lab_us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CavelloMember_8c797a38-da5d-4f08-9a45-bd2cf1b95121_terseLabel_en-US" xlink:label="lab_kg_CavelloMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cavello</link:label>
    <link:label id="lab_kg_CavelloMember_label_en-US" xlink:label="lab_kg_CavelloMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cavello [Member]</link:label>
    <link:label id="lab_kg_CavelloMember_documentation_en-US" xlink:label="lab_kg_CavelloMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Cavello</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CavelloMember" xlink:href="kg-20250930.xsd#kg_CavelloMember"/>
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    <link:label id="lab_ecd_AwardGrantDateFairValue_terseLabel_en-US" xlink:label="lab_ecd_AwardGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value as of Grant Date</link:label>
    <link:label id="lab_ecd_AwardGrantDateFairValue_label_en-US" xlink:label="lab_ecd_AwardGrantDateFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Award Grant Date Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AwardGrantDateFairValue" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AwardGrantDateFairValue"/>
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    <link:label id="lab_us-gaap_EquitySecuritiesFvNiCost_132b3680-6eb9-4879-a902-f5ebe2e9fb29_verboseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiCost" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Equity securities, at fair value, cost</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNiCost_dc7f671c-b4ad-42be-9978-dc2df3ab8711_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cost</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNiCost_label_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities, FV-NI, Cost</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquitySecuritiesFvNiCost" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquitySecuritiesFvNiCost"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquitySecuritiesFvNiCost" xlink:to="lab_us-gaap_EquitySecuritiesFvNiCost" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuableTreasurySharesRetained_74167437-6b81-4062-ad83-c9ea86107019_negatedTerseLabel_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuableTreasurySharesRetained" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Bermuda NewCo shares retained as treasury shares by Maiden Reinsurance at Closing (in shares)</link:label>
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    <link:label id="lab_us-gaap_OperatingLossCarryforwards_85ff7f72-5797-4765-bd85-0f59823555eb_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLossCarryforwards" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating loss carryforwards</link:label>
    <link:label id="lab_us-gaap_OperatingLossCarryforwards_label_en-US" xlink:label="lab_us-gaap_OperatingLossCarryforwards" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Loss Carryforwards</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLossCarryforwards" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLossCarryforwards"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLossCarryforwards" xlink:to="lab_us-gaap_OperatingLossCarryforwards" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock_33bdb387-ab42-4d3a-9610-170b831aa057_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Issuances of Senior Notes Outstanding</link:label>
    <link:label id="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfDebtInstrumentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Long-Term Debt Instruments [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfDebtInstrumentsTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfDebtInstrumentsTextBlock"/>
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    <link:label id="lab_us-gaap_RelatedPartyTransactionLineItems_28431088-c779-4ae2-bcf9-6235db571996_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Line Items]</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionLineItems_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionLineItems" xlink:to="lab_us-gaap_RelatedPartyTransactionLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal_63bfa71c-1c23-4b47-a2ed-54ce9f2a56ac_verboseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Expected credit losses on recoverables where credit losses were not previously recognized</link:label>
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    <link:label id="lab_us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable, Credit Loss Expense (Reversal)</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverableCreditLossExpenseReversal"/>
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    <link:label id="lab_dei_DocumentQuarterlyReport_d73649b9-83d1-44ae-a68d-fd3ebf84f40f_terseLabel_en-US" xlink:label="lab_dei_DocumentQuarterlyReport" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Document Quarterly Report</link:label>
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    <link:label id="lab_kg_SilverstoneVenture1Member_93bd3eaf-9b2b-4925-87f8-a743c974bed8_terseLabel_en-US" xlink:label="lab_kg_SilverstoneVenture1Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Silverstone Venture 1</link:label>
    <link:label id="lab_kg_SilverstoneVenture1Member_label_en-US" xlink:label="lab_kg_SilverstoneVenture1Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Silverstone Venture 1 [Member]</link:label>
    <link:label id="lab_kg_SilverstoneVenture1Member_documentation_en-US" xlink:label="lab_kg_SilverstoneVenture1Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Silverstone Venture 1</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_SilverstoneVenture1Member" xlink:href="kg-20250930.xsd#kg_SilverstoneVenture1Member"/>
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    <link:label id="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock_a95044a2-218a-47bd-8209-5bf8abef7ca8_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueAssetsAndLiabilitiesMeasuredOnRecurringAndNonrecurringBasisValuationTechniquesTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Significant Unobservable Inputs for Determining Fair Value of Other Investments</link:label>
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    <link:label id="lab_ecd_ChangedPeerGroupFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_ChangedPeerGroupFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Changed Peer Group, Footnote</link:label>
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    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_236347e8-a56e-42ec-8d83-7cdfd1add109_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2027</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Two</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearTwo"/>
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    <link:label id="lab_ecd_AdjToPeoCompFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_AdjToPeoCompFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustment To PEO Compensation, Footnote</link:label>
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    <link:label id="lab_us-gaap_AccruedInvestmentIncomeReceivable_2ec51b59-78b8-435e-a9f9-7abe98db52dd_terseLabel_en-US" xlink:label="lab_us-gaap_AccruedInvestmentIncomeReceivable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued investment income</link:label>
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    <link:label id="lab_us-gaap_InvestmentTypeAxis_e8c6c3e3-6bd7-48a3-a12c-7d149821cd02_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Type [Axis]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_Investments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_Investments"/>
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    <link:label id="lab_kg_GainLossOnOtherSecuritiesExcludingDebtAndEquitySecurities_084d7901-f669-410b-be74-f370b9fbd061_totalLabel_en-US" xlink:label="lab_kg_GainLossOnOtherSecuritiesExcludingDebtAndEquitySecurities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Other investments, net</link:label>
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    <link:label id="lab_kg_ScheduleOfInvestmentCreditRatingsTableTextBlock_fc6399c0-e393-4f90-b55e-739df8193d7e_terseLabel_en-US" xlink:label="lab_kg_ScheduleOfInvestmentCreditRatingsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of the Credit Ratings of Fixed Maturities</link:label>
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    <link:label id="lab_kg_LiabilityforUnpaidClaimsandClaimsAdjustmentExpensePeriodIncreaseDecreaseNetofCommissionChanges_6ddfa518-1ae5-428c-9c9f-3e9ce02fac8f_negatedLabel_en-US" xlink:label="lab_kg_LiabilityforUnpaidClaimsandClaimsAdjustmentExpensePeriodIncreaseDecreaseNetofCommissionChanges" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Total PPD for Legacy Reinsurance Segment</link:label>
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    <link:label id="lab_us-gaap_PaymentsToAcquireBusinessesGross_5be5602f-64ab-4cf7-8f06-0aa75cc6d254_terseLabel_en-US" xlink:label="lab_us-gaap_PaymentsToAcquireBusinessesGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Upfront cash</link:label>
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    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleAllowanceForCreditLoss_1ea85bcf-6500-433c-a922-bb202eda5ae8_terseLabel_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleAllowanceForCreditLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Allowance for expected credit losses</link:label>
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    <link:label id="lab_us-gaap_FeeIncome_48058d20-7203-4be6-ac12-8cd31ea082f8_terseLabel_en-US" xlink:label="lab_us-gaap_FeeIncome" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fee revenue</link:label>
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    <link:label id="lab_kg_GenesisLegacySolutionsMember_39cf3244-977c-41ad-a913-6f070d9545c5_terseLabel_en-US" xlink:label="lab_kg_GenesisLegacySolutionsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Genesis Legacy Solutions</link:label>
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    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossLineItems_deb041b2-5fb7-42cd-9ab0-1583c2634856_terseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated Other Comprehensive Income (Loss) [Line Items]</link:label>
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    <link:label id="lab_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedGainBeforeTax_7ad574e8-90c1-45e2-b225-9d6a3cbb0fc5_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleDebtSecuritiesAccumulatedGrossUnrealizedGainBeforeTax" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gross unrealized gains</link:label>
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    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_817e7fa0-1b4b-4d1a-9929-fb02bea9181e_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2026</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year One</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueNextTwelveMonths"/>
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    <link:label id="lab_us-gaap_PaymentsForProceedsFromLoansReceivable_d77a8e6f-cbfd-490d-88d9-23fd5b82a016_negatedLabel_en-US" xlink:label="lab_us-gaap_PaymentsForProceedsFromLoansReceivable" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Net loan receivable from related party</link:label>
    <link:label id="lab_us-gaap_PaymentsForProceedsFromLoansReceivable_label_en-US" xlink:label="lab_us-gaap_PaymentsForProceedsFromLoansReceivable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payments for (Proceeds from) Loans Receivable</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsForProceedsFromLoansReceivable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsForProceedsFromLoansReceivable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PaymentsForProceedsFromLoansReceivable" xlink:to="lab_us-gaap_PaymentsForProceedsFromLoansReceivable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims_40ae7e4b-8fdc-4b1f-9a0b-fbd982a94e3f_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional claims estimated</link:label>
    <link:label id="lab_us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims_label_en-US" xlink:label="lab_us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverables, Incurred but Not Reported Claims</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceReceivablesIncurredButNotReportedClaims"/>
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    <link:label id="lab_ecd_EqtyAwrdsAdjsMember_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments</link:label>
    <link:label id="lab_ecd_EqtyAwrdsAdjsMember_label_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_EqtyAwrdsAdjsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_EqtyAwrdsAdjsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_EqtyAwrdsAdjsMember" xlink:to="lab_ecd_EqtyAwrdsAdjsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PnsnBnftsAdjFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_PnsnBnftsAdjFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pension Benefits Adjustments, Footnote</link:label>
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    <link:label id="lab_kg_RestrictedCashAndInvestmentsLineItems_fbe08569-e389-41ed-be5c-29cb4c4bea17_terseLabel_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments Items [Line Items]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsLineItems_label_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments [Line Items]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsLineItems_documentation_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RestrictedCashAndInvestmentsLineItems" xlink:href="kg-20250930.xsd#kg_RestrictedCashAndInvestmentsLineItems"/>
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    <link:label id="lab_kg_NumberOfDirectors_5f03b4b6-8467-4f47-b4e5-a36e94a75707_terseLabel_en-US" xlink:label="lab_kg_NumberOfDirectors" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of directors</link:label>
    <link:label id="lab_kg_NumberOfDirectors_label_en-US" xlink:label="lab_kg_NumberOfDirectors" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number Of Directors</link:label>
    <link:label id="lab_kg_NumberOfDirectors_documentation_en-US" xlink:label="lab_kg_NumberOfDirectors" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number Of Directors</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_NumberOfDirectors" xlink:href="kg-20250930.xsd#kg_NumberOfDirectors"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_NumberOfDirectors" xlink:to="lab_kg_NumberOfDirectors" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_OutstandingRecoveryCompAmt_terseLabel_en-US" xlink:label="lab_ecd_OutstandingRecoveryCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Compensation Amount</link:label>
    <link:label id="lab_ecd_OutstandingRecoveryCompAmt_label_en-US" xlink:label="lab_ecd_OutstandingRecoveryCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Recovery Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingRecoveryCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingRecoveryCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OutstandingRecoveryCompAmt" xlink:to="lab_ecd_OutstandingRecoveryCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_SeniorNotes2013Member_2315a140-07e8-4b9d-a83c-826d1641a95e_verboseLabel_en-US" xlink:label="lab_kg_SeniorNotes2013Member" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Senior Notes - MHNC &#8211; 7.75%</link:label>
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    <link:label id="lab_kg_SeniorNotes2013Member_documentation_en-US" xlink:label="lab_kg_SeniorNotes2013Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Senior Notes 2013 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_SeniorNotes2013Member" xlink:href="kg-20250930.xsd#kg_SeniorNotes2013Member"/>
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    <link:label id="lab_us-gaap_DebtInstrumentNameDomain_f41c992a-bdea-49f6-8b5c-e4764028e120_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentNameDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument, Name [Domain]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentNameDomain_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentNameDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentNameDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentNameDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentNameDomain" xlink:to="lab_us-gaap_DebtInstrumentNameDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfLiabilityForUnpaidClaimsAndClaimsAdjustmentExpense_004e26ce-54d1-4dd1-a68c-fd56a9d285ff_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfLiabilityForUnpaidClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule for Reserve for Loss and LAE and Schedule of Prior Period Development Experienced</link:label>
    <link:label id="lab_us-gaap_ScheduleOfLiabilityForUnpaidClaimsAndClaimsAdjustmentExpense_label_en-US" xlink:label="lab_us-gaap_ScheduleOfLiabilityForUnpaidClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Liability for Unpaid Claims and Claims Adjustment Expense [Table Text Block]</link:label>
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    <link:label id="lab_ecd_RecoveryOfErrCompDisclosureLineItems_label_en-US" xlink:label="lab_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Recovery of Erroneously Awarded Compensation Disclosure [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_RecoveryOfErrCompDisclosureLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:to="lab_ecd_RecoveryOfErrCompDisclosureLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember_308c7b6a-3185-4c87-af0e-44b1194d4e30_terseLabel_en-US" xlink:label="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Estimate of Fair Value Measurement</link:label>
    <link:label id="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember_5c0380fd-5a40-48e5-8460-926615f022f0_verboseLabel_en-US" xlink:label="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair Value</link:label>
    <link:label id="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember_label_en-US" xlink:label="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Estimate of Fair Value Measurement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EstimateOfFairValueFairValueDisclosureMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:to="lab_us-gaap_EstimateOfFairValueFairValueDisclosureMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal_a8517a09-efc6-46e8-b194-d13c7c1016f3_verboseLabel_en-US" xlink:label="lab_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">% of Total</link:label>
    <link:label id="lab_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal_label_en-US" xlink:label="lab_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Investments, Entities That Calculate Net Asset Value Per Share, Unfunded Commitments, Percentage Of Total</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal" xlink:href="kg-20250930.xsd#kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal" xlink:to="lab_kg_FairValueInvestmentsEntitiesThatCalculateNetAssetValuePerShareUnfundedCommitmentsPercentageOfTotal" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod_e20caf23-13cd-4b2f-bfc8-1144df125e92_terseLabel_en-US" xlink:label="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Policy inception, grace period</link:label>
    <link:label id="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod_label_en-US" xlink:label="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insurance Contracts, Program Fee Receivables, Premium Receivable, Collection Period</link:label>
    <link:label id="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod_documentation_en-US" xlink:label="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Insurance Contracts, Program Fee Receivables, Premium Receivable, Collection Period</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:href="kg-20250930.xsd#kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:to="lab_kg_InsuranceContractsProgramFeeReceivablesPremiumReceivableCollectionPeriod" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfInvestmentsLineItems_53330eb7-7187-48a6-bf1d-4bc0f0896eb1_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfInvestmentsLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Investments [Line Items]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfInvestmentsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfInvestmentsLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfInvestmentsLineItems" xlink:to="lab_us-gaap_ScheduleOfInvestmentsLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_MnpiDiscTimedForCompValFlag_terseLabel_en-US" xlink:label="lab_ecd_MnpiDiscTimedForCompValFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">MNPI Disclosure Timed for Compensation Value</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_MnpiDiscTimedForCompValFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_MnpiDiscTimedForCompValFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_MnpiDiscTimedForCompValFlag" xlink:to="lab_ecd_MnpiDiscTimedForCompValFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsAmortizedCost_5141d47c-13f0-4d82-9f55-9ece7331cede_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterOneThroughFiveYearsAmortizedCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Due after one year through five years</link:label>
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    <link:label id="lab_us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList_93f98f16-c2c4-4d92-a548-8054be4db65d_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating Lease, Right-of-Use Asset, Statement of Financial Position [Extensible Enumeration]</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Right-of-Use Asset, Statement of Financial Position [Extensible Enumeration]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseRightOfUseAssetStatementOfFinancialPositionExtensibleList"/>
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    <link:label id="lab_us-gaap_IncomeTaxDisclosureTextBlock_fa72cfa6-992e-4dcf-aa81-d22d3d20261b_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Income Taxes</link:label>
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    <link:label id="lab_kg_InvestmentsUnrealizedGainLossGrossOfDeferredIncomeTaxExpense_ebad5b60-eb32-40a6-be00-c60254971154_terseLabel_en-US" xlink:label="lab_kg_InvestmentsUnrealizedGainLossGrossOfDeferredIncomeTaxExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total net unrealized losses</link:label>
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    <link:label id="lab_ecd_OutstandingRecoveryIndName_terseLabel_en-US" xlink:label="lab_ecd_OutstandingRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_OutstandingRecoveryIndName_label_en-US" xlink:label="lab_ecd_OutstandingRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Outstanding Recovery, Individual Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_OutstandingRecoveryIndName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_OutstandingRecoveryIndName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_OutstandingRecoveryIndName" xlink:to="lab_ecd_OutstandingRecoveryIndName" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty_d7ac9b5a-7b5e-4f91-aaba-2bd386f8a6d8_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percent of losses with respect to current lines of business, excluding those above covered business threshold to related party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Losses With Respect to Current Lines of Business, Excluding Those Above Covered Business Threshold To Related Party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent of Losses With Respect to Current Lines of Business, Excluding Those Above Covered Business Threshold To Related Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:to="lab_kg_RelatedPartyTransactionPercentOfLossesWithRespectToCurrentLinesOfBusinessExcludingThoseAboveCoveredBusinessThresholdToRelatedParty" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AdditionalPaidInCapital_aba0fc81-d8bf-448c-93e7-cbd6b61b64ea_terseLabel_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapital" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Additional paid-in capital</link:label>
    <link:label id="lab_us-gaap_AdditionalPaidInCapital_label_en-US" xlink:label="lab_us-gaap_AdditionalPaidInCapital" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Additional Paid in Capital</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AdditionalPaidInCapital" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AdditionalPaidInCapital"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AdditionalPaidInCapital" xlink:to="lab_us-gaap_AdditionalPaidInCapital" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ProductOrServiceAxis_86fc660e-8c01-42b5-a30f-63f107374f92_terseLabel_en-US" xlink:label="lab_srt_ProductOrServiceAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Product and Service [Axis]</link:label>
    <link:label id="lab_srt_ProductOrServiceAxis_label_en-US" xlink:label="lab_srt_ProductOrServiceAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Product and Service [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ProductOrServiceAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ProductOrServiceAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ProductOrServiceAxis" xlink:to="lab_srt_ProductOrServiceAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements_5dc7a6e0-2c02-44e5-8be5-f935471b8c05_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Funds withheld receivable</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Funds Held under Reinsurance Agreements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements" xlink:to="lab_us-gaap_IncreaseDecreaseInFundsHeldUnderReinsuranceAgreements" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable_918eb3b8-5e3b-4ee6-becb-a39f61c2a7e6_terseLabel_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination, Intangible Asset, Acquired, Finite-Lived [Table]</link:label>
    <link:label id="lab_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable_label_en-US" xlink:label="lab_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Intangible Asset, Acquired, Finite-Lived [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable" xlink:to="lab_us-gaap_FiniteLivedIntangibleAssetsAcquiredAsPartOfBusinessCombinationTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross_60139e0b-987f-4c4b-940d-0685eb238056_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverables on unpaid losses</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverables on Unpaid Losses, Gross</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross" xlink:to="lab_us-gaap_ReinsuranceRecoverablesOnUnpaidLossesGross" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CededCreditRiskReinsurerDomain_b282d676-98e4-4e03-99bd-217b0e8efb91_terseLabel_en-US" xlink:label="lab_us-gaap_CededCreditRiskReinsurerDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurer, Name [Domain]</link:label>
    <link:label id="lab_us-gaap_CededCreditRiskReinsurerDomain_label_en-US" xlink:label="lab_us-gaap_CededCreditRiskReinsurerDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurer, Name [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CededCreditRiskReinsurerDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CededCreditRiskReinsurerDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CededCreditRiskReinsurerDomain" xlink:to="lab_us-gaap_CededCreditRiskReinsurerDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RestrictedInvestmentLiability_0e1c5955-1f28-407b-9f1f-5269a90de122_negatedTerseLabel_en-US" xlink:label="lab_kg_RestrictedInvestmentLiability" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Restricted investments &#8211; liability for investments purchased for related party agreements</link:label>
    <link:label id="lab_kg_RestrictedInvestmentLiability_label_en-US" xlink:label="lab_kg_RestrictedInvestmentLiability" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Investment, Liability</link:label>
    <link:label id="lab_kg_RestrictedInvestmentLiability_documentation_en-US" xlink:label="lab_kg_RestrictedInvestmentLiability" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Restricted Investment, Liability</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RestrictedInvestmentLiability" xlink:href="kg-20250930.xsd#kg_RestrictedInvestmentLiability"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RestrictedInvestmentLiability" xlink:to="lab_kg_RestrictedInvestmentLiability" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses_8d82c019-a433-493b-9c1c-7103d272e233_negatedTerseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Reserve for loss and loss adjustment expenses</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses_label_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Reserve For Loss And Loss Adjustment Expenses</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses_documentation_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination Recognized Identifiable Assets Acquired And Liabilities Assumed, Reserve For Loss And Loss Adjustment Expenses</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:to="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedReserveForLossAndLossAdjustmentExpenses" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AssumedPremiumsEarned_1adcf21f-fbc1-46f6-92da-22b6437ed070_terseLabel_en-US" xlink:label="lab_us-gaap_AssumedPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Assumed</link:label>
    <link:label id="lab_us-gaap_AssumedPremiumsEarned_label_en-US" xlink:label="lab_us-gaap_AssumedPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assumed Premiums Earned</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AssumedPremiumsEarned" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssumedPremiumsEarned"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AssumedPremiumsEarned" xlink:to="lab_us-gaap_AssumedPremiumsEarned" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_RangeAxis_86237458-4a1d-48a2-864b-b2cf62607eb6_terseLabel_en-US" xlink:label="lab_srt_RangeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Axis]</link:label>
    <link:label id="lab_srt_RangeAxis_label_en-US" xlink:label="lab_srt_RangeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_RangeAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_RangeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_RangeAxis" xlink:to="lab_srt_RangeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FundsWithheldArrangementMember_65d76667-b589-4419-8aa9-3c972c894a41_terseLabel_en-US" xlink:label="lab_kg_FundsWithheldArrangementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Income on funds withheld</link:label>
    <link:label id="lab_kg_FundsWithheldArrangementMember_label_en-US" xlink:label="lab_kg_FundsWithheldArrangementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Funds Withheld Arrangement [Member]</link:label>
    <link:label id="lab_kg_FundsWithheldArrangementMember_documentation_en-US" xlink:label="lab_kg_FundsWithheldArrangementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Funds Withheld Arrangement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FundsWithheldArrangementMember" xlink:href="kg-20250930.xsd#kg_FundsWithheldArrangementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FundsWithheldArrangementMember" xlink:to="lab_kg_FundsWithheldArrangementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PrivateCreditLendingMember_8dbdaf0f-bcfd-4eb5-a961-4ae70adfce03_terseLabel_en-US" xlink:label="lab_kg_PrivateCreditLendingMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Private credit investments</link:label>
    <link:label id="lab_kg_PrivateCreditLendingMember_label_en-US" xlink:label="lab_kg_PrivateCreditLendingMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Private Credit Lending [Member]</link:label>
    <link:label id="lab_kg_PrivateCreditLendingMember_documentation_en-US" xlink:label="lab_kg_PrivateCreditLendingMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Private Credit Lending</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PrivateCreditLendingMember" xlink:href="kg-20250930.xsd#kg_PrivateCreditLendingMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PrivateCreditLendingMember" xlink:to="lab_kg_PrivateCreditLendingMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PremiumRepaymentLoanAgreementMember_9d23f4ac-efbc-413b-ba82-a875289e800c_terseLabel_en-US" xlink:label="lab_kg_PremiumRepaymentLoanAgreementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Premium Repayment Loan Agreement</link:label>
    <link:label id="lab_kg_PremiumRepaymentLoanAgreementMember_label_en-US" xlink:label="lab_kg_PremiumRepaymentLoanAgreementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premium Repayment Loan Agreement [Member]</link:label>
    <link:label id="lab_kg_PremiumRepaymentLoanAgreementMember_documentation_en-US" xlink:label="lab_kg_PremiumRepaymentLoanAgreementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Premium Repayment Loan Agreement</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PremiumRepaymentLoanAgreementMember" xlink:href="kg-20250930.xsd#kg_PremiumRepaymentLoanAgreementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PremiumRepaymentLoanAgreementMember" xlink:to="lab_kg_PremiumRepaymentLoanAgreementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_MarketableSecuritiesRealizedGain_5e70fdcb-76ba-4d39-9924-fb5f8909c7d4_terseLabel_en-US" xlink:label="lab_kg_MarketableSecuritiesRealizedGain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other investments, gross gains</link:label>
    <link:label id="lab_kg_MarketableSecuritiesRealizedGain_label_en-US" xlink:label="lab_kg_MarketableSecuritiesRealizedGain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Marketable Securities, Realized Gain</link:label>
    <link:label id="lab_kg_MarketableSecuritiesRealizedGain_documentation_en-US" xlink:label="lab_kg_MarketableSecuritiesRealizedGain" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Marketable Securities, Realized Gain</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_MarketableSecuritiesRealizedGain" xlink:href="kg-20250930.xsd#kg_MarketableSecuritiesRealizedGain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_MarketableSecuritiesRealizedGain" xlink:to="lab_kg_MarketableSecuritiesRealizedGain" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AggtErrCompAmt_terseLabel_en-US" xlink:label="lab_ecd_AggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Erroneous Compensation Amount</link:label>
    <link:label id="lab_ecd_AggtErrCompAmt_label_en-US" xlink:label="lab_ecd_AggtErrCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Aggregate Erroneous Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AggtErrCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AggtErrCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AggtErrCompAmt" xlink:to="lab_ecd_AggtErrCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching_57501438-f063-4bc7-8ea6-29ffaa363f15_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Maximum limit of liability attaching</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Maximum Limit of Liability Attaching</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Maximum Limit of Liability Attaching</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:to="lab_kg_RelatedPartyTransactionMaximumLimitOfLiabilityAttaching" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeerGroupIssuersFnTextBlock_terseLabel_en-US" xlink:label="lab_ecd_PeerGroupIssuersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Peer Group Issuers, Footnote</link:label>
    <link:label id="lab_ecd_PeerGroupIssuersFnTextBlock_label_en-US" xlink:label="lab_ecd_PeerGroupIssuersFnTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Peer Group Issuers, Footnote [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeerGroupIssuersFnTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeerGroupIssuersFnTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeerGroupIssuersFnTextBlock" xlink:to="lab_ecd_PeerGroupIssuersFnTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_392065ae-b250-4ea1-b8cb-6093a3118b19_totalLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net (loss) income attributable to Kestrel common shareholders, basic</link:label>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic_label_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Net Income (Loss) Available to Common Stockholders, Basic</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:to="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasic" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentOwnershipAtMergerClosingMember_f8a8bb92-a349-4cea-8596-68d8a9860938_terseLabel_en-US" xlink:label="lab_kg_InvestmentOwnershipAtMergerClosingMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Ownership, Issued at closing of Combination</link:label>
    <link:label id="lab_kg_InvestmentOwnershipAtMergerClosingMember_label_en-US" xlink:label="lab_kg_InvestmentOwnershipAtMergerClosingMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Ownership, At Merger Closing [Member]</link:label>
    <link:label id="lab_kg_InvestmentOwnershipAtMergerClosingMember_documentation_en-US" xlink:label="lab_kg_InvestmentOwnershipAtMergerClosingMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investment Ownership, At Merger Closing</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentOwnershipAtMergerClosingMember" xlink:href="kg-20250930.xsd#kg_InvestmentOwnershipAtMergerClosingMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentOwnershipAtMergerClosingMember" xlink:to="lab_kg_InvestmentOwnershipAtMergerClosingMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_b855e2a7-f279-4213-a74c-605db688a52f_totalLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Other comprehensive loss, after tax</link:label>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_96b4dd5e-e2f4-49bb-8912-b374c58878d4_terseLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net current period other comprehensive income (loss)</link:label>
    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent_label_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Comprehensive Income (Loss), Net of Tax, Portion Attributable to Parent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:to="lab_us-gaap_OtherComprehensiveIncomeLossNetOfTaxPortionAttributableToParent" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour_01844e7d-fe65-41dc-b6c3-f037fdbe8cc7_terseLabel_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Thereafter</link:label>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour_label_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, Payments Due After Year Four</link:label>
    <link:label id="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour_documentation_en-US" xlink:label="lab_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, Payments Due After Year Four</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour" xlink:href="kg-20250930.xsd#kg_LesseeOperatingLeaseLiabilityPaymentsDueAfterYearFour"/>
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    <link:label id="lab_us-gaap_OperatingExpenses_c3cd2409-7b8a-4ce0-bed4-8b0fc5ce2786_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_OperatingExpenses" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">General and administrative expenses</link:label>
    <link:label id="lab_us-gaap_OperatingExpenses_label_en-US" xlink:label="lab_us-gaap_OperatingExpenses" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Expenses</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingExpenses" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingExpenses"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingExpenses" xlink:to="lab_us-gaap_OperatingExpenses" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessAcquisitionSharePrice_968cda04-f9ec-4ca2-898f-53ec9968aeb4_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionSharePrice" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Bermuda NewCo share price (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_BusinessAcquisitionSharePrice_1f7afa8d-56c4-4cd9-a3dc-50a380ecb7d6_verboseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionSharePrice" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Bermuda NewCo share price (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_BusinessAcquisitionSharePrice_label_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionSharePrice" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Price Per Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessAcquisitionSharePrice" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessAcquisitionSharePrice"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessAcquisitionSharePrice" xlink:to="lab_us-gaap_BusinessAcquisitionSharePrice" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ReinsuranceAgreementCompletingRunOff_34250fbb-3bfa-4c13-a81a-c6f791e334c2_terseLabel_en-US" xlink:label="lab_kg_ReinsuranceAgreementCompletingRunOff" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of reinsurance agreements, completing run-off</link:label>
    <link:label id="lab_kg_ReinsuranceAgreementCompletingRunOff_label_en-US" xlink:label="lab_kg_ReinsuranceAgreementCompletingRunOff" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Agreement, Completing Run-Off</link:label>
    <link:label id="lab_kg_ReinsuranceAgreementCompletingRunOff_documentation_en-US" xlink:label="lab_kg_ReinsuranceAgreementCompletingRunOff" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Reinsurance Agreement, Completing Run-Off</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ReinsuranceAgreementCompletingRunOff" xlink:href="kg-20250930.xsd#kg_ReinsuranceAgreementCompletingRunOff"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ReinsuranceAgreementCompletingRunOff" xlink:to="lab_kg_ReinsuranceAgreementCompletingRunOff" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueInputsLevel1Member_3bff9a90-b93e-496a-815a-bca863664dee_verboseLabel_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel1Member" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Quoted Prices in Active Markets for Identical Assets (Level 1)</link:label>
    <link:label id="lab_us-gaap_FairValueInputsLevel1Member_label_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel1Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Inputs, Level 1 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueInputsLevel1Member" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueInputsLevel1Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueInputsLevel1Member" xlink:to="lab_us-gaap_FairValueInputsLevel1Member" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_StandardPoorsAAARatingMember_df44fe85-c618-4798-bac4-7c9ab8a32324_terseLabel_en-US" xlink:label="lab_srt_StandardPoorsAAARatingMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">AAA</link:label>
    <link:label id="lab_srt_StandardPoorsAAARatingMember_label_en-US" xlink:label="lab_srt_StandardPoorsAAARatingMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Standard &amp; Poor's, AAA Rating [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_StandardPoorsAAARatingMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_StandardPoorsAAARatingMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_StandardPoorsAAARatingMember" xlink:to="lab_srt_StandardPoorsAAARatingMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_e42d0b47-ec91-40e7-b9a2-c07193bd8b1c_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Related Party Transactions, by Related Party [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:to="lab_us-gaap_ScheduleOfRelatedPartyTransactionsByRelatedPartyTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EarningsPerShareDiluted_35148603-ddd4-4f21-b8ad-5ea65381dd66_totalLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareDiluted" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Diluted (loss) income per share attributable to Kestrel common shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareDiluted_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareDiluted" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Diluted</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EarningsPerShareDiluted" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EarningsPerShareDiluted"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EarningsPerShareDiluted" xlink:to="lab_us-gaap_EarningsPerShareDiluted" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_SilverstoneVenture3Member_25593d7e-82ff-4cc6-86f5-42c7507bca34_terseLabel_en-US" xlink:label="lab_kg_SilverstoneVenture3Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Silverstone Venture 3</link:label>
    <link:label id="lab_kg_SilverstoneVenture3Member_label_en-US" xlink:label="lab_kg_SilverstoneVenture3Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Silverstone Venture 3 [Member]</link:label>
    <link:label id="lab_kg_SilverstoneVenture3Member_documentation_en-US" xlink:label="lab_kg_SilverstoneVenture3Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Silverstone Venture 3</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_SilverstoneVenture3Member" xlink:href="kg-20250930.xsd#kg_SilverstoneVenture3Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_SilverstoneVenture3Member" xlink:to="lab_kg_SilverstoneVenture3Member" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense_42694c0e-167e-42b7-8db9-c95b099370d0_terseLabel_en-US" xlink:label="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net unrealized gains, net of deferred income tax</link:label>
    <link:label id="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense_label_en-US" xlink:label="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investments, Gross Unrealized Gain (Loss), Net of Deferred Income Tax Expense</link:label>
    <link:label id="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense_documentation_en-US" xlink:label="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investments, Gross Unrealized Gain (Loss), Net of Deferred Income Tax Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:href="kg-20250930.xsd#kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:to="lab_kg_InvestmentsGrossUnrealizedGainLossNetOfDeferredIncomeTaxExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue_22f43b55-642d-4070-8451-9672af286ce9_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Due after five years through ten years</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Fair Value, Maturity, Allocated and Single Maturity Date, after Year 5 Through 10</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAfterFiveThroughTenYearsFairValue"/>
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    <link:label id="lab_kg_ScheduleofFixedMaturitiesTableLineItems_8a223f5b-24e8-407a-a47a-4a2e71c52dad_terseLabel_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTableLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Fixed Maturities Table [Line Items]</link:label>
    <link:label id="lab_kg_ScheduleofFixedMaturitiesTableLineItems_label_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTableLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Fixed Maturities Table [Line Items]</link:label>
    <link:label id="lab_kg_ScheduleofFixedMaturitiesTableLineItems_documentation_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTableLineItems" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">[Line Items] for Schedule of Fixed Maturities Table [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ScheduleofFixedMaturitiesTableLineItems" xlink:href="kg-20250930.xsd#kg_ScheduleofFixedMaturitiesTableLineItems"/>
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    <link:label id="lab_us-gaap_ReinsuranceAccountingPolicy_a34d7f25-c95f-4660-a4de-a7b0d8030d02_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceAccountingPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance and Retroactive Reinsurance</link:label>
    <link:label id="lab_us-gaap_ReinsuranceAccountingPolicy_label_en-US" xlink:label="lab_us-gaap_ReinsuranceAccountingPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Accounting Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceAccountingPolicy" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceAccountingPolicy"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceAccountingPolicy" xlink:to="lab_us-gaap_ReinsuranceAccountingPolicy" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_e3352107-0166-4eab-bee4-0dc845a280ec_terseLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Treasury shares, at cost (in shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_8a7e20db-a35b-4540-899e-0dd573107fc6_netLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Treasury shares (in shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonShares_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonShares" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common, Shares</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonShares" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonShares"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TreasuryStockCommonShares" xlink:to="lab_us-gaap_TreasuryStockCommonShares" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_KestrelIntermediateLedbetterHoldingsMember_75a57ec7-53a7-4a49-bba2-6d0934148857_terseLabel_en-US" xlink:label="lab_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Kestrel Intermediate Ledbetter Holdings</link:label>
    <link:label id="lab_kg_KestrelIntermediateLedbetterHoldingsMember_label_en-US" xlink:label="lab_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Kestrel Intermediate Ledbetter Holdings [Member]</link:label>
    <link:label id="lab_kg_KestrelIntermediateLedbetterHoldingsMember_documentation_en-US" xlink:label="lab_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Kestrel Intermediate Ledbetter Holdings</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:href="kg-20250930.xsd#kg_KestrelIntermediateLedbetterHoldingsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:to="lab_kg_KestrelIntermediateLedbetterHoldingsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ConsolidationItemsDomain_5e1296ed-ca28-4d3d-b43a-a39c2471d426_terseLabel_en-US" xlink:label="lab_srt_ConsolidationItemsDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Consolidation Items [Domain]</link:label>
    <link:label id="lab_srt_ConsolidationItemsDomain_label_en-US" xlink:label="lab_srt_ConsolidationItemsDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Consolidation Items [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ConsolidationItemsDomain" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ConsolidationItemsDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ConsolidationItemsDomain" xlink:to="lab_srt_ConsolidationItemsDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CededCreditRiskAxis_97d98624-9867-4dbc-a230-80db54a2347d_terseLabel_en-US" xlink:label="lab_us-gaap_CededCreditRiskAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurer, Name [Axis]</link:label>
    <link:label id="lab_us-gaap_CededCreditRiskAxis_label_en-US" xlink:label="lab_us-gaap_CededCreditRiskAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurer, Name [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CededCreditRiskAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CededCreditRiskAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CededCreditRiskAxis" xlink:to="lab_us-gaap_CededCreditRiskAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRecoverables_db701938-4708-41a6-bd82-dc861b8cd7a8_periodStartLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverables" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Less: reinsurance recoverable on unpaid losses, January 1</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverables_1e493ebe-bda3-4c30-abd3-5912cd7378e7_periodEndLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverables" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverable on unpaid losses, September 30</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverables_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverables" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverables, Including Reinsurance Premium Paid</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverables" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverables"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRecoverables" xlink:to="lab_us-gaap_ReinsuranceRecoverables" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems_49b324b0-8a77-4ea5-9219-9305fbac75f7_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Gain (Loss) on Securities [Line Items]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems_label_en-US" xlink:label="lab_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain (Loss) on Securities [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems" xlink:to="lab_us-gaap_ScheduleOfGainLossOnInvestmentsIncludingMarketableSecuritiesAndInvestmentsHeldAtCostIncomeStatementReportedAmountsSummaryLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_HistoricReinsuranceMember_4b02e0bc-5229-4480-8b8b-839d1973b3d3_terseLabel_en-US" xlink:label="lab_kg_HistoricReinsuranceMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Historic Reinsurance</link:label>
    <link:label id="lab_kg_HistoricReinsuranceMember_label_en-US" xlink:label="lab_kg_HistoricReinsuranceMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Historic Reinsurance [Member]</link:label>
    <link:label id="lab_kg_HistoricReinsuranceMember_documentation_en-US" xlink:label="lab_kg_HistoricReinsuranceMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Historic Reinsurance</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_HistoricReinsuranceMember" xlink:href="kg-20250930.xsd#kg_HistoricReinsuranceMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_HistoricReinsuranceMember" xlink:to="lab_kg_HistoricReinsuranceMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit_61def7e7-45f0-40f6-bae3-f593642bd702_terseLabel_en-US" xlink:label="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percentage of outstanding debt holders in lawsuit (at least)</link:label>
    <link:label id="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit_label_en-US" xlink:label="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loss Contingency, Percentage of Outstanding Debt Holders Involved in Lawsuit</link:label>
    <link:label id="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit_documentation_en-US" xlink:label="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Loss Contingency, Percentage of Outstanding Debt Holders Involved in Lawsuit</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:href="kg-20250930.xsd#kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:to="lab_kg_LossContingencyPercentageOfOutstandingDebtHoldersInvolvedInLawsuit" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CorporateBondSecuritiesMember_a6f189c4-4fcf-49b6-9d72-d3f758574c45_terseLabel_en-US" xlink:label="lab_us-gaap_CorporateBondSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Corporate bonds</link:label>
    <link:label id="lab_us-gaap_CorporateBondSecuritiesMember_label_en-US" xlink:label="lab_us-gaap_CorporateBondSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Corporate Bond Securities [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CorporateBondSecuritiesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CorporateBondSecuritiesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CorporateBondSecuritiesMember" xlink:to="lab_us-gaap_CorporateBondSecuritiesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentMilestoneTypeAxis_d61bdba6-c4fd-436f-a1f4-9643c2f12658_terseLabel_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type [Axis]</link:label>
    <link:label id="lab_kg_InvestmentMilestoneTypeAxis_label_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type [Axis]</link:label>
    <link:label id="lab_kg_InvestmentMilestoneTypeAxis_documentation_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeAxis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentMilestoneTypeAxis" xlink:href="kg-20250930.xsd#kg_InvestmentMilestoneTypeAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentMilestoneTypeAxis" xlink:to="lab_kg_InvestmentMilestoneTypeAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PostTerminationEndorsementNo.1Member_dd524c87-976b-4ddc-83bc-f5a122e920f0_terseLabel_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.1Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 1</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementNo.1Member_label_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.1Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 1 [Member]</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementNo.1Member_documentation_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.1Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 1 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PostTerminationEndorsementNo.1Member" xlink:href="kg-20250930.xsd#kg_PostTerminationEndorsementNo.1Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PostTerminationEndorsementNo.1Member" xlink:to="lab_kg_PostTerminationEndorsementNo.1Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentCarryingAmount_8fd85f9b-c3ac-4b0d-9025-c19af9a6bc27_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentCarryingAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Senior notes - principal amount</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentCarryingAmount_e2b2ca4e-62bb-46e5-b03d-cf42f972b856_verboseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentCarryingAmount" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Principal amount</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentCarryingAmount_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentCarryingAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Gross</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentCarryingAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentCarryingAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentCarryingAmount" xlink:to="lab_us-gaap_DebtInstrumentCarryingAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PostTerminationEndorsementMember_d87cb865-d4da-4bb8-933f-efb0680113c3_terseLabel_en-US" xlink:label="lab_kg_PostTerminationEndorsementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementMember_label_en-US" xlink:label="lab_kg_PostTerminationEndorsementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement [Member]</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementMember_documentation_en-US" xlink:label="lab_kg_PostTerminationEndorsementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PostTerminationEndorsementMember" xlink:href="kg-20250930.xsd#kg_PostTerminationEndorsementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PostTerminationEndorsementMember" xlink:to="lab_kg_PostTerminationEndorsementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InvestmentOwnedBalanceShares_3b75b646-b3e4-4029-9d5d-4390cb4334f6_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentOwnedBalanceShares" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock owned by shareholder (in shares)</link:label>
    <link:label id="lab_us-gaap_InvestmentOwnedBalanceShares_label_en-US" xlink:label="lab_us-gaap_InvestmentOwnedBalanceShares" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Owned, Balance, Shares</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentOwnedBalanceShares" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentOwnedBalanceShares"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentOwnedBalanceShares" xlink:to="lab_us-gaap_InvestmentOwnedBalanceShares" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrExpirationDate_terseLabel_en-US" xlink:label="lab_ecd_TrdArrExpirationDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expiration Date</link:label>
    <link:label id="lab_ecd_TrdArrExpirationDate_label_en-US" xlink:label="lab_ecd_TrdArrExpirationDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Expiration Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrExpirationDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrExpirationDate"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrExpirationDate" xlink:to="lab_ecd_TrdArrExpirationDate" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RelatedPartyTransactionAmountsOfTransaction_e642019f-35a5-49db-8288-6c14683028af_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party transaction amount (returned)</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionAmountsOfTransaction_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Amounts of Transaction</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionAmountsOfTransaction"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:to="lab_us-gaap_RelatedPartyTransactionAmountsOfTransaction" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract_aee88231-e195-4257-ac97-dc82f61b668b_terseLabel_en-US" xlink:label="lab_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">EQUITY</link:label>
    <link:label id="lab_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract_label_en-US" xlink:label="lab_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity, Including Portion Attributable to Noncontrolling Interest [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract" xlink:to="lab_us-gaap_StockholdersEquityIncludingPortionAttributableToNoncontrollingInterestAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted_d526af13-567a-4044-8104-e35267cabf50_terseLabel_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative arrangement, reserves expected to be commuted</link:label>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted_label_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Reserves Expected to be Commuted</link:label>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted_documentation_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommuted" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Reserves Expected to be Commuted</link:label>
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    <link:label id="lab_us-gaap_EquitySecuritiesFvNiUnrealizedGainLoss_c7dfde37-b3a6-40b1-a114-d8ababcc3cd9_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiUnrealizedGainLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net unrealized gains recognized for equity securities still held at the reporting date</link:label>
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    <link:label id="lab_us-gaap_FixedIncomeSecuritiesMember_fe6cacaa-d6fe-446c-acfb-67c092882320_terseLabel_en-US" xlink:label="lab_us-gaap_FixedIncomeSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fixed maturities</link:label>
    <link:label id="lab_us-gaap_FixedIncomeSecuritiesMember_label_en-US" xlink:label="lab_us-gaap_FixedIncomeSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Income Securities [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FixedIncomeSecuritiesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FixedIncomeSecuritiesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FixedIncomeSecuritiesMember" xlink:to="lab_us-gaap_FixedIncomeSecuritiesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherInvestmentsMember_1a1649b9-f44e-47cd-8f1e-1ae5404967c0_terseLabel_en-US" xlink:label="lab_us-gaap_OtherInvestmentsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other equity method investments at fair value option elected</link:label>
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    <link:label id="lab_us-gaap_EquityMethodInvestmentsTextBlock_ebde041c-6c54-4d36-b23d-7d342d93661c_terseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Equity Method Investments and Carrying Value and Beneficial Ownership Percentage of Equity Method Investments</link:label>
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    <link:label id="lab_kg_ShareCapitalSharesAuthorized_668f5cda-89f4-4bff-9353-391c9d5bfe79_terseLabel_en-US" xlink:label="lab_kg_ShareCapitalSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate shares authorized (in shares)</link:label>
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    <link:label id="lab_kg_StandardPoorsSingleaPlusSingleaAndSingleaMinusMember_fc9cf347-d5dd-443a-818f-04fb0d277f48_terseLabel_en-US" xlink:label="lab_kg_StandardPoorsSingleaPlusSingleaAndSingleaMinusMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">A+, A, A-</link:label>
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    <link:label id="lab_srt_ConsolidatedEntitiesDomain_8383aa38-83ba-4bc2-b42b-9f89730aff03_terseLabel_en-US" xlink:label="lab_srt_ConsolidatedEntitiesDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Consolidated Entities [Domain]</link:label>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearAmortizedCost_9b891f63-bcd2-41df-abc2-0aebbfdd6b89_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearAmortizedCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Due in one year or less</link:label>
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    <link:label id="lab_us-gaap_ReinsuranceRecoverablesOnPaidLossesGross_4917c299-f4b2-4f08-8aa8-0ad566e2601f_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesOnPaidLossesGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverables on paid losses, gross</link:label>
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    <link:label id="lab_kg_NumberOfIndependentDirectors_4d2b7fb4-6d13-4b7a-ad2b-57acdce37f6e_terseLabel_en-US" xlink:label="lab_kg_NumberOfIndependentDirectors" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of independent directors</link:label>
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    <link:label id="lab_us-gaap_FinancialInstrumentAxis_91a6621a-518f-45f9-96bc-efd34dceff27_terseLabel_en-US" xlink:label="lab_us-gaap_FinancialInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Financial Instrument [Axis]</link:label>
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    <link:label id="lab_us-gaap_BenefitsLossesAndExpensesAbstract_54a08fa5-ddf9-4b07-aa49-f885cbb5a270_terseLabel_en-US" xlink:label="lab_us-gaap_BenefitsLossesAndExpensesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Expenses</link:label>
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    <link:label id="lab_us-gaap_DerivativeLiabilities_7054af7f-677e-4104-9c16-dfdc737b8e7d_terseLabel_en-US" xlink:label="lab_us-gaap_DerivativeLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Underwriting-related derivative liability</link:label>
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    <link:label id="lab_ecd_EqtyAwrdsAdjsExclgValRprtdInSummryCompstnTblMember_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsAdjsExclgValRprtdInSummryCompstnTblMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity Awards Adjustments, Excluding Value Reported in Compensation Table</link:label>
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    <link:label id="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilitiesHeldForSale_261c9167-5af7-4949-aba4-de9d1890c4cb_negatedTerseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedLiabilitiesHeldForSale" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Liabilities held for sale</link:label>
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    <link:label id="lab_kg_InvestmentOwnership5PercentMember_076a19ca-d2bd-4350-9763-66d9790989cb_terseLabel_en-US" xlink:label="lab_kg_InvestmentOwnership5PercentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Ownership, 5%</link:label>
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    <link:label id="lab_us-gaap_StockIssuedDuringPeriodValueAcquisitions_28a81b5a-a655-4d85-807d-e3a4ddb748b6_terseLabel_en-US" xlink:label="lab_us-gaap_StockIssuedDuringPeriodValueAcquisitions" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Shares issued on Combination Agreement</link:label>
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    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNet_e32411fb-6c9f-4f57-9e66-ff55065de3d8_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net assets</link:label>
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    <link:label id="lab_kg_InvestmentOwnership25PercentMember_6edb0d71-9813-4844-993c-e38062fdb7ca_terseLabel_en-US" xlink:label="lab_kg_InvestmentOwnership25PercentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Ownership, 25%</link:label>
    <link:label id="lab_kg_InvestmentOwnership25PercentMember_label_en-US" xlink:label="lab_kg_InvestmentOwnership25PercentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Ownership, 25 Percent [Member]</link:label>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities_6bf296d6-7702-46ac-8514-c19a4e34027b_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash used in financing activities</link:label>
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    <link:label id="lab_ecd_MeasureAxis_terseLabel_en-US" xlink:label="lab_ecd_MeasureAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Measure:</link:label>
    <link:label id="lab_ecd_MeasureAxis_label_en-US" xlink:label="lab_ecd_MeasureAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measure [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_MeasureAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_MeasureAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_MeasureAxis" xlink:to="lab_ecd_MeasureAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_MaidenHoldingsLtd.Member_2e607575-5218-480a-985d-2c66bbf4d4c1_terseLabel_en-US" xlink:label="lab_kg_MaidenHoldingsLtd.Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Maiden Holdings, Ltd.</link:label>
    <link:label id="lab_kg_MaidenHoldingsLtd.Member_label_en-US" xlink:label="lab_kg_MaidenHoldingsLtd.Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Maiden Holdings, Ltd. [Member]</link:label>
    <link:label id="lab_kg_MaidenHoldingsLtd.Member_documentation_en-US" xlink:label="lab_kg_MaidenHoldingsLtd.Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Maiden Holdings, Ltd. [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_MaidenHoldingsLtd.Member" xlink:href="kg-20250930.xsd#kg_MaidenHoldingsLtd.Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_MaidenHoldingsLtd.Member" xlink:to="lab_kg_MaidenHoldingsLtd.Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ValuationTechniqueAxis_fc1173e1-65e2-4629-ab74-f0e73a82ce72_terseLabel_en-US" xlink:label="lab_us-gaap_ValuationTechniqueAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Valuation Approach and Technique [Axis]</link:label>
    <link:label id="lab_us-gaap_ValuationTechniqueAxis_label_en-US" xlink:label="lab_us-gaap_ValuationTechniqueAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Valuation Approach and Technique [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ValuationTechniqueAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ValuationTechniqueAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ValuationTechniqueAxis" xlink:to="lab_us-gaap_ValuationTechniqueAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_SeidenLPSeidenMGMTLPMember_dda73c91-d5a9-48a5-901b-2cd0bab51365_terseLabel_en-US" xlink:label="lab_kg_SeidenLPSeidenMGMTLPMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Seiden LP &amp; Seiden MGMT LP</link:label>
    <link:label id="lab_kg_SeidenLPSeidenMGMTLPMember_label_en-US" xlink:label="lab_kg_SeidenLPSeidenMGMTLPMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Seiden LP &amp; Seiden MGMT LP [Member]</link:label>
    <link:label id="lab_kg_SeidenLPSeidenMGMTLPMember_documentation_en-US" xlink:label="lab_kg_SeidenLPSeidenMGMTLPMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Seiden LP &amp; Seiden MGMT LP</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_SeidenLPSeidenMGMTLPMember" xlink:href="kg-20250930.xsd#kg_SeidenLPSeidenMGMTLPMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_SeidenLPSeidenMGMTLPMember" xlink:to="lab_kg_SeidenLPSeidenMGMTLPMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupClassificationDomain_5d965517-109c-442e-8d9d-28146a125bc4_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupClassificationDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disposal Group Classification [Domain]</link:label>
    <link:label id="lab_us-gaap_DisposalGroupClassificationDomain_label_en-US" xlink:label="lab_us-gaap_DisposalGroupClassificationDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group Classification [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupClassificationDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupClassificationDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupClassificationDomain" xlink:to="lab_us-gaap_DisposalGroupClassificationDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BenefitsLossesAndExpensesNet_96333e45-a070-496d-ac82-9fd1d081b89f_totalLabel_en-US" xlink:label="lab_kg_BenefitsLossesAndExpensesNet" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total expenses</link:label>
    <link:label id="lab_kg_BenefitsLossesAndExpensesNet_label_en-US" xlink:label="lab_kg_BenefitsLossesAndExpensesNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Benefits, Losses and Expenses, Net</link:label>
    <link:label id="lab_kg_BenefitsLossesAndExpensesNet_documentation_en-US" xlink:label="lab_kg_BenefitsLossesAndExpensesNet" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Benefits, Losses and Expenses, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BenefitsLossesAndExpensesNet" xlink:href="kg-20250930.xsd#kg_BenefitsLossesAndExpensesNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BenefitsLossesAndExpensesNet" xlink:to="lab_kg_BenefitsLossesAndExpensesNet" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PvpTableTextBlock_terseLabel_en-US" xlink:label="lab_ecd_PvpTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure, Table</link:label>
    <link:label id="lab_ecd_PvpTableTextBlock_label_en-US" xlink:label="lab_ecd_PvpTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pay vs Performance [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PvpTableTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PvpTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PvpTableTextBlock" xlink:to="lab_ecd_PvpTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Violation of Home Country Law, Amount</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery due to Violation of Home Country Law, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:to="lab_ecd_ForgoneRecoveryDueToViolationOfHomeCountryLawAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AmortizationOfFinancingCosts_4a58884b-b719-428a-9175-54d75df6d6cb_terseLabel_en-US" xlink:label="lab_us-gaap_AmortizationOfFinancingCosts" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortization for the fair value adjustment</link:label>
    <link:label id="lab_us-gaap_AmortizationOfFinancingCosts_label_en-US" xlink:label="lab_us-gaap_AmortizationOfFinancingCosts" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Amortization of Debt Issuance Costs</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AmortizationOfFinancingCosts" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AmortizationOfFinancingCosts"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AmortizationOfFinancingCosts" xlink:to="lab_us-gaap_AmortizationOfFinancingCosts" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss_4acf9ec1-ab77-45fe-8224-1e49ef8c318e_terseLabel_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unrealized losses of unrealized loss positions</link:label>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss_label_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investments, Continuous Unrealized Loss Position, Accumulated Loss</link:label>
    <link:label id="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss_documentation_en-US" xlink:label="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investments, Continuous Unrealized Loss Position, Accumulated Loss</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:href="kg-20250930.xsd#kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:to="lab_kg_InvestmentsContinuousUnrealizedLossPositionAccumulatedLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InvestmentIncomeInvestmentExpense_cfa9e2a5-33a5-495a-8194-84595e655b39_negatedLabel_en-US" xlink:label="lab_us-gaap_InvestmentIncomeInvestmentExpense" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Investment expenses</link:label>
    <link:label id="lab_us-gaap_InvestmentIncomeInvestmentExpense_label_en-US" xlink:label="lab_us-gaap_InvestmentIncomeInvestmentExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Income, Investment Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentIncomeInvestmentExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentIncomeInvestmentExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentIncomeInvestmentExpense" xlink:to="lab_us-gaap_InvestmentIncomeInvestmentExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityTaxIdentificationNumber_a2d398e9-7dc3-4657-bafd-be7a7d1ea248_terseLabel_en-US" xlink:label="lab_dei_EntityTaxIdentificationNumber" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Tax Identification Number</link:label>
    <link:label id="lab_dei_EntityTaxIdentificationNumber_label_en-US" xlink:label="lab_dei_EntityTaxIdentificationNumber" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Tax Identification Number</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityTaxIdentificationNumber" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityTaxIdentificationNumber"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityTaxIdentificationNumber" xlink:to="lab_dei_EntityTaxIdentificationNumber" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ValuationTechniqueDiscountedCashFlowMember_80a9bcbf-b612-4fc7-9430-4ef4df163b47_terseLabel_en-US" xlink:label="lab_us-gaap_ValuationTechniqueDiscountedCashFlowMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discounted cash flows</link:label>
    <link:label id="lab_us-gaap_ValuationTechniqueDiscountedCashFlowMember_label_en-US" xlink:label="lab_us-gaap_ValuationTechniqueDiscountedCashFlowMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Valuation Technique, Discounted Cash Flow [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ValuationTechniqueDiscountedCashFlowMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ValuationTechniqueDiscountedCashFlowMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ValuationTechniqueDiscountedCashFlowMember" xlink:to="lab_us-gaap_ValuationTechniqueDiscountedCashFlowMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue_95b38b3f-d2dc-4e67-9036-d4af9f97db70_verboseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair value of securities without single maturities</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Maturity, without Single Maturity Date, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue" xlink:to="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithoutSingleMaturityDateFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves_a0ad3861-a972-4be6-88a1-99932a2e6b56_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair value discount on acquired net reserves for losses and LAE</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves_label_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Recognized Asset Acquired, Identifiable Intangible Asset, Fair Value Discount on Acquired Reserves</link:label>
    <link:label id="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves_documentation_en-US" xlink:label="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Recognized Asset Acquired, Identifiable Intangible Asset, Fair Value Discount on Acquired Reserves</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:href="kg-20250930.xsd#kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:to="lab_kg_BusinessCombinationRecognizedAssetAcquiredIdentifiableIntangibleAssetFairValueDiscountOnAcquiredReserves" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeDomain_457c9de4-c8fa-4e27-8926-58bb2450c406_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Domain]</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeDomain_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Domain]</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionByTypeDomain_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionByTypeDomain" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, By Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionByTypeDomain" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionByTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionByTypeDomain" xlink:to="lab_kg_RelatedPartyTransactionByTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockSharesIssued_d708f324-8054-4f5c-b892-ca67a423f650_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockSharesIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares, issued (in shares)</link:label>
    <link:label id="lab_us-gaap_CommonStockSharesIssued_label_en-US" xlink:label="lab_us-gaap_CommonStockSharesIssued" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Shares, Issued</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CommonStockSharesIssued" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CommonStockSharesIssued"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CommonStockSharesIssued" xlink:to="lab_us-gaap_CommonStockSharesIssued" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_35f4d786-56ca-41ee-97fb-c6446f867212_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating lease, weighted average remaining lease term</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1_label_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Operating Lease, Weighted Average Remaining Lease Term</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:to="lab_us-gaap_OperatingLeaseWeightedAverageRemainingLeaseTerm1" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover_68ba9cd3-404e-42d5-936f-cdda4f381082_terseLabel_en-US" xlink:label="lab_kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverable for paid and unpaid claims and claims adjustments, adverse development cover</link:label>
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    <link:label id="lab_kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover_documentation_en-US" xlink:label="lab_kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable For Paid And Unpaid Claims And Claims Adjustments, Adverse Development Cover</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover" xlink:href="kg-20250930.xsd#kg_ReinsuranceRecoverableForPaidAndUnpaidClaimsAndClaimsAdjustmentsAdverseDevelopmentCover"/>
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    <link:label id="lab_us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember_7e2a2c76-fc77-4527-918a-be940b224745_terseLabel_en-US" xlink:label="lab_us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">U.S. agency bonds&#160;&#8211;&#160;mortgage-backed</link:label>
    <link:label id="lab_us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember_label_en-US" xlink:label="lab_us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Mortgage-Backed Security, Issued by US Government-Sponsored Enterprise [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MortgageBackedSecuritiesIssuedByUSGovernmentSponsoredEnterprisesMember"/>
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    <link:label id="lab_ecd_TradingArrAxis_terseLabel_en-US" xlink:label="lab_ecd_TradingArrAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Arrangement:</link:label>
    <link:label id="lab_ecd_TradingArrAxis_label_en-US" xlink:label="lab_ecd_TradingArrAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TradingArrAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TradingArrAxis"/>
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    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract_9ce1dcec-8114-4226-8c82-774d4d5f160d_terseLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unrealized losses</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract_label_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Accumulated Loss [Abstract]</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Accumulated Loss [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:to="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionAccumulatedLossAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TotalShareholderRtnAmt_terseLabel_en-US" xlink:label="lab_ecd_TotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Amount</link:label>
    <link:label id="lab_ecd_TotalShareholderRtnAmt_label_en-US" xlink:label="lab_ecd_TotalShareholderRtnAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Total Shareholder Return Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TotalShareholderRtnAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TotalShareholderRtnAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TotalShareholderRtnAmt" xlink:to="lab_ecd_TotalShareholderRtnAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentMilestoneTypeDomain_f1596844-9395-4aa0-b1d0-37383c7eae38_terseLabel_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type [Domain]</link:label>
    <link:label id="lab_kg_InvestmentMilestoneTypeDomain_label_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type [Domain]</link:label>
    <link:label id="lab_kg_InvestmentMilestoneTypeDomain_documentation_en-US" xlink:label="lab_kg_InvestmentMilestoneTypeDomain" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investment Milestone, Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentMilestoneTypeDomain" xlink:href="kg-20250930.xsd#kg_InvestmentMilestoneTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentMilestoneTypeDomain" xlink:to="lab_kg_InvestmentMilestoneTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PostTerminationEndorsementNo.2Member_55b994a3-f8db-4b28-bc0c-fd9b1e95600c_terseLabel_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.2Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 2</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementNo.2Member_label_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.2Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 2 [Member]</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementNo.2Member_documentation_en-US" xlink:label="lab_kg_PostTerminationEndorsementNo.2Member" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement No. 2 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PostTerminationEndorsementNo.2Member" xlink:href="kg-20250930.xsd#kg_PostTerminationEndorsementNo.2Member"/>
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    <link:label id="lab_us-gaap_CorporateNonSegmentMember_e68810e5-a607-44ee-8431-778838e19c67_verboseLabel_en-US" xlink:label="lab_us-gaap_CorporateNonSegmentMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Corporate assets</link:label>
    <link:label id="lab_us-gaap_CorporateNonSegmentMember_label_en-US" xlink:label="lab_us-gaap_CorporateNonSegmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Segment Reporting, Reconciling Item, Corporate Nonsegment [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CorporateNonSegmentMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CorporateNonSegmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CorporateNonSegmentMember" xlink:to="lab_us-gaap_CorporateNonSegmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_SecurityExchangeName_eca577e0-5657-4fa6-a174-ac7c0b80b28b_terseLabel_en-US" xlink:label="lab_dei_SecurityExchangeName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Security Exchange Name</link:label>
    <link:label id="lab_dei_SecurityExchangeName_label_en-US" xlink:label="lab_dei_SecurityExchangeName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Security Exchange Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_SecurityExchangeName" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_SecurityExchangeName"/>
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    <link:label id="lab_us-gaap_ReinsuranceRecoverablesAllowance_9d8628b8-efb3-481e-b442-319188c6331b_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesAllowance" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance recoverable, allowance for credit loss</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverablesAllowance_ade8a903-666c-4cb3-a78e-8240926d1623_periodStartLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesAllowance" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Allowance for expected credit losses on reinsurance recoverable, beginning of period</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverablesAllowance_c3906e3f-2aae-48a2-962b-3071da4bc5ae_periodEndLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesAllowance" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Allowance for expected credit losses on reinsurance recoverable, end of period</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverablesAllowance_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverablesAllowance" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable, Allowance for Credit Loss</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverablesAllowance" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverablesAllowance"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRecoverablesAllowance" xlink:to="lab_us-gaap_ReinsuranceRecoverablesAllowance" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeAxis_a7df9ca3-0e4a-4e8e-a763-7f6ac11505bf_terseLabel_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Axis]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeAxis_label_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Axis]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeAxis_documentation_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeAxis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RestrictedCashAndInvestmentsTypeAxis" xlink:href="kg-20250930.xsd#kg_RestrictedCashAndInvestmentsTypeAxis"/>
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    <link:label id="lab_us-gaap_CededPremiumsEarned_e95295da-15e6-4fb7-b09e-2f3cf1a12e88_terseLabel_en-US" xlink:label="lab_us-gaap_CededPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Ceded</link:label>
    <link:label id="lab_us-gaap_CededPremiumsEarned_label_en-US" xlink:label="lab_us-gaap_CededPremiumsEarned" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Ceded Premiums Earned</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CededPremiumsEarned" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CededPremiumsEarned"/>
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    <link:label id="lab_us-gaap_InvestmentHoldingsTextBlock_dae5505d-0b59-40ee-9b51-b3b528f5d745_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentHoldingsTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investments</link:label>
    <link:label id="lab_us-gaap_InvestmentHoldingsTextBlock_label_en-US" xlink:label="lab_us-gaap_InvestmentHoldingsTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Holdings [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentHoldingsTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentHoldingsTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentHoldingsTextBlock" xlink:to="lab_us-gaap_InvestmentHoldingsTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceDisclosuresAbstract_9aa9591e-d6b9-4b5a-a2e8-09278622c5ed_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceDisclosuresAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance Disclosures [Abstract]</link:label>
    <link:label id="lab_us-gaap_ReinsuranceDisclosuresAbstract_label_en-US" xlink:label="lab_us-gaap_ReinsuranceDisclosuresAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Disclosures [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceDisclosuresAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceDisclosuresAbstract"/>
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    <link:label id="lab_us-gaap_StatementLineItems_fe67228a-3886-481b-8155-91f4e27fd906_terseLabel_en-US" xlink:label="lab_us-gaap_StatementLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement [Line Items]</link:label>
    <link:label id="lab_us-gaap_StatementLineItems_label_en-US" xlink:label="lab_us-gaap_StatementLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementLineItems"/>
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    <link:label id="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue_015fa937-79a3-42c0-be91-d133985d8e2d_terseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue_fa3d22e2-b459-4413-b3c2-eba68f1a5133_verboseLabel_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents, end of period</link:label>
    <link:label id="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue_label_en-US" xlink:label="lab_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash and Cash Equivalent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashAndCashEquivalentsAtCarryingValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashAndCashEquivalentsAtCarryingValue"/>
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    <link:label id="lab_us-gaap_CommonStockMember_5917987c-7772-4361-8756-d12b0023bd18_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares</link:label>
    <link:label id="lab_us-gaap_CommonStockMember_fdcf310c-c0a4-4a8a-99f0-60d7cf6aba8c_verboseLabel_en-US" xlink:label="lab_us-gaap_CommonStockMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Privately held common stocks</link:label>
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    <link:label id="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies_ee953035-ff9a-4934-a456-ea6d3ede0317_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Premiums and losses related to policies, percentage</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Premiums and Losses Related to Policies</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Premiums and Losses Related to Policies</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies" xlink:to="lab_kg_RelatedPartyTransactionPremiumsAndLossesRelatedToPolicies" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_RangeMember_e31e1781-634e-4a46-9413-09f5d52365a0_terseLabel_en-US" xlink:label="lab_srt_RangeMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statistical Measurement [Domain]</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_srt_RangeMember" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_RangeMember"/>
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    <link:label id="lab_kg_RelatedPartyTransactionPercentageOfRequiredFundingOnObligations_6a653180-45a0-4df6-a498-6b8a27ce17f6_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentageOfRequiredFundingOnObligations" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related party transaction, percentage of required funding on obligations</link:label>
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    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleAllowanceForCreditLossWriteoff_abe52566-5788-451b-b7a2-64a4ca51714d_terseLabel_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleAllowanceForCreditLossWriteoff" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Allowance for expected credit losses</link:label>
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    <link:label id="lab_us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersPriorYearClaimsAndClaimsAdjustmentExpense_00eea4fc-c29f-4d73-9fe1-b12dc244d1f4_terseLabel_en-US" xlink:label="lab_us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersPriorYearClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net incurred losses related to prior years</link:label>
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    <link:label id="lab_us-gaap_RelatedPartyTransactionAxis_ca120e20-1d23-403a-90ff-5ceef6c3ffb2_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transaction [Axis]</link:label>
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    <link:label id="lab_us-gaap_UnearnedPremiums_8ceac1b8-c1ae-4e8f-bd6b-2aa1b62cfeb9_verboseLabel_en-US" xlink:label="lab_us-gaap_UnearnedPremiums" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Unearned premiums (includes $19,401 from related parties in 2025)</link:label>
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    <link:label id="lab_us-gaap_AcquiredFiniteLivedIntangibleAssetsLineItems_label_en-US" xlink:label="lab_us-gaap_AcquiredFiniteLivedIntangibleAssetsLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Intangible Asset, Acquired, Finite-Lived [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AcquiredFiniteLivedIntangibleAssetsLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AcquiredFiniteLivedIntangibleAssetsLineItems"/>
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    <link:label id="lab_us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued_b2955283-be19-4a80-96ed-c2036acb0469_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionEquityInterestsIssuedOrIssuableNumberOfSharesIssued" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares (in shares)</link:label>
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    <link:label id="lab_us-gaap_GainContingenciesLineItems_label_en-US" xlink:label="lab_us-gaap_GainContingenciesLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain Contingencies [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_GainContingenciesLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_GainContingenciesLineItems"/>
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    <link:label id="lab_ecd_TradingArrByIndTable_terseLabel_en-US" xlink:label="lab_ecd_TradingArrByIndTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Trading Arrangements, by Individual</link:label>
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    <link:label id="lab_us-gaap_EarningsPerShareBasic_fde7336d-d4ae-44d9-80dc-6568e7df0d21_totalLabel_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasic" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Basic (loss) income per share attributable to Kestrel common shareholders (in dollars per share)</link:label>
    <link:label id="lab_us-gaap_EarningsPerShareBasic_label_en-US" xlink:label="lab_us-gaap_EarningsPerShareBasic" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Earnings Per Share, Basic</link:label>
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    <link:label id="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRollForward_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable, Allowance for Credit Loss [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRollForward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverableAllowanceForCreditLossRollForward"/>
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    <link:label id="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock_a17c8b4e-a189-4632-ac19-e9bf782e2306_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfSegmentReportingInformationBySegmentTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Reconciliation of Reportable Segments Underwriting Results to Consolidated Net Loss</link:label>
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    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerDilutedShare_966e581f-78cb-4b67-aa02-d9ae6abd3415_verboseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxPerDilutedShare" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Diluted loss from discontinued operations per share attributable to Kestrel common shareholders (in dollars per share)</link:label>
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    <link:label id="lab_kg_InterestAndAmortizationExpensesNet_a86d9a00-0b36-4eba-a36e-08b77088b9f8_terseLabel_en-US" xlink:label="lab_kg_InterestAndAmortizationExpensesNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest and amortization expenses</link:label>
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    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionTwelveMonthsorLongerFairValue_fc4e2a55-9e04-4111-81bb-a29eafd187fc_terseLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionTwelveMonthsorLongerFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">12 Months or More</link:label>
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    <link:label id="lab_us-gaap_TreasuryStockCommonMember_9dc5ca47-e680-4c9e-9df1-3da11880f658_terseLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Treasury shares</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonMember_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonMember"/>
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    <link:label id="lab_ecd_PayVsPerformanceDisclosureLineItems_label_en-US" xlink:label="lab_ecd_PayVsPerformanceDisclosureLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PayVsPerformanceDisclosureLineItems" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PayVsPerformanceDisclosureLineItems"/>
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    <link:label id="lab_us-gaap_OperatingLeaseLiability_6bad41ef-5e50-48df-9519-85072ec14d8f_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiability" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating lease, liability</link:label>
    <link:label id="lab_us-gaap_OperatingLeaseLiability_43ca1497-056d-4ebf-8f62-59b580a9e3d8_totalLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiability" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total discounted operating lease liabilities</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OperatingLeaseLiability" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OperatingLeaseLiability"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OperatingLeaseLiability" xlink:to="lab_us-gaap_OperatingLeaseLiability" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_USGovernmentAgenciesDebtSecuritiesMember_8ae0bb46-c508-4b98-9b26-c514d723a38c_terseLabel_en-US" xlink:label="lab_us-gaap_USGovernmentAgenciesDebtSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">U.S. agency bonds&#160;&#8211;&#160;mortgage-backed</link:label>
    <link:label id="lab_us-gaap_USGovernmentAgenciesDebtSecuritiesMember_label_en-US" xlink:label="lab_us-gaap_USGovernmentAgenciesDebtSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">US Government Agencies Debt Securities [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_USGovernmentAgenciesDebtSecuritiesMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_USGovernmentAgenciesDebtSecuritiesMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_USGovernmentAgenciesDebtSecuritiesMember" xlink:to="lab_us-gaap_USGovernmentAgenciesDebtSecuritiesMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_IncreaseDecreaseInUnearnedPremiums_b17cc6cb-d22b-4986-b766-3f5f6e695c0c_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInUnearnedPremiums" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Unearned premiums</link:label>
    <link:label id="lab_us-gaap_IncreaseDecreaseInUnearnedPremiums_label_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInUnearnedPremiums" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Increase (Decrease) in Unearned Premiums</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncreaseDecreaseInUnearnedPremiums" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncreaseDecreaseInUnearnedPremiums"/>
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    <link:label id="lab_us-gaap_LiabilitiesAbstract_2b59bc1c-3fd8-4e55-ab82-b09b4308793b_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">LIABILITIES</link:label>
    <link:label id="lab_us-gaap_LiabilitiesAbstract_label_en-US" xlink:label="lab_us-gaap_LiabilitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liabilities [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesAbstract" xlink:to="lab_us-gaap_LiabilitiesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationDisclosureTextBlock_97c68ab2-da05-43b8-9e56-b3a9fa18f599_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationDisclosureTextBlock" xlink:to="lab_us-gaap_BusinessCombinationDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FinancialLiabilitiesFairValueDisclosure_e9ca18e8-7aed-4e5b-a1ad-da67ca724cf5_terseLabel_en-US" xlink:label="lab_us-gaap_FinancialLiabilitiesFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total Senior Notes</link:label>
    <link:label id="lab_us-gaap_FinancialLiabilitiesFairValueDisclosure_label_en-US" xlink:label="lab_us-gaap_FinancialLiabilitiesFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Financial Liabilities Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancialLiabilitiesFairValueDisclosure" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancialLiabilitiesFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FinancialLiabilitiesFairValueDisclosure" xlink:to="lab_us-gaap_FinancialLiabilitiesFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember_00136ccd-c255-4a6b-88c5-17277c391718_terseLabel_en-US" xlink:label="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Quarterly financial statements</link:label>
    <link:label id="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember_label_en-US" xlink:label="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Valuation Technique, Quarterly Financial Statements [Member]</link:label>
    <link:label id="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember_documentation_en-US" xlink:label="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Valuation Technique, Quarterly Financial Statements</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:href="kg-20250930.xsd#kg_ValuationTechniqueQuarterlyFinancialStatementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:to="lab_kg_ValuationTechniqueQuarterlyFinancialStatementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ProgramServicesSegmentMember_9a868a72-b135-4eb3-a43f-5b4579ed1274_terseLabel_en-US" xlink:label="lab_kg_ProgramServicesSegmentMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Program Services</link:label>
    <link:label id="lab_kg_ProgramServicesSegmentMember_label_en-US" xlink:label="lab_kg_ProgramServicesSegmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Program Services Segment [Member]</link:label>
    <link:label id="lab_kg_ProgramServicesSegmentMember_documentation_en-US" xlink:label="lab_kg_ProgramServicesSegmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Program Services Segment</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ProgramServicesSegmentMember" xlink:href="kg-20250930.xsd#kg_ProgramServicesSegmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ProgramServicesSegmentMember" xlink:to="lab_kg_ProgramServicesSegmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves_d6e4d963-82f9-4f1e-a1ca-74ca0d87bd80_terseLabel_en-US" xlink:label="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative arrangement, payment of commuted reserves</link:label>
    <link:label id="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves_label_en-US" xlink:label="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Payment of Commuted Reserves</link:label>
    <link:label id="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves_documentation_en-US" xlink:label="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Payment of Commuted Reserves</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:href="kg-20250930.xsd#kg_CollaborativeArrangementPaymentOfCommutedReserves"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:to="lab_kg_CollaborativeArrangementPaymentOfCommutedReserves" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_515e0cf8-a30f-46e2-aa56-c0a12b5fcfba_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value, Assets Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</link:label>
    <link:label id="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward_label_en-US" xlink:label="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Assets Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward"/>
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    <link:label id="lab_us-gaap_ScheduleOfVariableInterestEntitiesTable_24b12736-ce72-4369-9f1a-91f37265023c_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfVariableInterestEntitiesTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Variable Interest Entities [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfVariableInterestEntitiesTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfVariableInterestEntitiesTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Variable Interest Entity [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfVariableInterestEntitiesTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfVariableInterestEntitiesTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfVariableInterestEntitiesTable" xlink:to="lab_us-gaap_ScheduleOfVariableInterestEntitiesTable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfCashFlowsAbstract_010f1d54-2625-4b59-806d-84a3c0f8e73a_terseLabel_en-US" xlink:label="lab_us-gaap_StatementOfCashFlowsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement of Cash Flows [Abstract]</link:label>
    <link:label id="lab_us-gaap_StatementOfCashFlowsAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfCashFlowsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Cash Flows [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfCashFlowsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfCashFlowsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfCashFlowsAbstract" xlink:to="lab_us-gaap_StatementOfCashFlowsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_CoSelectedMeasureAmt_terseLabel_en-US" xlink:label="lab_ecd_CoSelectedMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Company Selected Measure Amount</link:label>
    <link:label id="lab_ecd_CoSelectedMeasureAmt_label_en-US" xlink:label="lab_ecd_CoSelectedMeasureAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Company Selected Measure Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_CoSelectedMeasureAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_CoSelectedMeasureAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_CoSelectedMeasureAmt" xlink:to="lab_ecd_CoSelectedMeasureAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_854f8684-3d71-48d8-ab2e-b6efa735fb4c_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2028</link:label>
    <link:label id="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree_label_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Lessee, Operating Lease, Liability, to be Paid, Year Three</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:to="lab_us-gaap_LesseeOperatingLeaseLiabilityPaymentsDueYearThree" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DeferredFinanceCostsGross_acb61fbf-a1ea-4e0d-a883-12794729d795_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredFinanceCostsGross" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Original fair value adjustment at acquisition date</link:label>
    <link:label id="lab_us-gaap_DeferredFinanceCostsGross_label_en-US" xlink:label="lab_us-gaap_DeferredFinanceCostsGross" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Issuance Costs, Gross</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredFinanceCostsGross" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredFinanceCostsGross"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredFinanceCostsGross" xlink:to="lab_us-gaap_DeferredFinanceCostsGross" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain_dadbb65e-bf9f-47b4-bc6b-77555b90099f_terseLabel_en-US" xlink:label="lab_kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity securities, gross gains</link:label>
    <link:label id="lab_kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain_label_en-US" xlink:label="lab_kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities, FV-NI, Realized And Unrealized Gain</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain" xlink:href="kg-20250930.xsd#kg_EquitySecuritiesFVNIRealizedAndUnrealizedGain"/>
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    <link:label id="lab_us-gaap_ComprehensiveIncomeNetOfTax_a18d5cdc-3965-4e3f-8983-3243881b21d8_totalLabel_en-US" xlink:label="lab_us-gaap_ComprehensiveIncomeNetOfTax" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Comprehensive (loss) income</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ComprehensiveIncomeNetOfTax" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ComprehensiveIncomeNetOfTax"/>
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    <link:label id="lab_us-gaap_MeasurementInputTypeAxis_label_en-US" xlink:label="lab_us-gaap_MeasurementInputTypeAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Measurement Input Type [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_MeasurementInputTypeAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_MeasurementInputTypeAxis"/>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue_2ad58f9b-57d8-4d3d-8138-49a92e009a8c_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Due in one year or less</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesWithinOneYearFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Fair Value, Maturity, Allocated and Single Maturity Date, Year One</link:label>
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    <link:label id="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParent_b49d9cee-6466-4f5b-ba96-1464d8a0b7f1_totalLabel_en-US" xlink:label="lab_us-gaap_OtherComprehensiveIncomeLossBeforeTaxPortionAttributableToParent" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Other comprehensive loss, before tax</link:label>
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    <link:label id="lab_us-gaap_ClassOfStockLineItems_f6e73ee4-089a-4fd1-b75c-4d755bbf7199_terseLabel_en-US" xlink:label="lab_us-gaap_ClassOfStockLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Class of Stock [Line Items]</link:label>
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    <link:label id="lab_kg_InvestmentsOriginalOrAmortizedCostBasis_0326cc41-2354-44cb-a059-772af9cd7337_terseLabel_en-US" xlink:label="lab_kg_InvestmentsOriginalOrAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortized cost</link:label>
    <link:label id="lab_kg_InvestmentsOriginalOrAmortizedCostBasis_label_en-US" xlink:label="lab_kg_InvestmentsOriginalOrAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investments, Original or Amortized Cost Basis</link:label>
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    <link:label id="lab_kg_InvestmentsFairValueOptionMember_5acdca81-4e46-4658-8885-34e0d41f345f_terseLabel_en-US" xlink:label="lab_kg_InvestmentsFairValueOptionMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total equity method investments with fair value option (included in Other Investments)</link:label>
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    <link:label id="lab_kg_InvestmentsFairValueOptionMember_documentation_en-US" xlink:label="lab_kg_InvestmentsFairValueOptionMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investments, Fair Value Option</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentsFairValueOptionMember" xlink:href="kg-20250930.xsd#kg_InvestmentsFairValueOptionMember"/>
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    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxAttributableToReportingEntity_092d3912-5ecf-41fa-81d2-ebb05164fb29_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxAttributableToReportingEntity" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Loss from discontinued operations, net of income tax</link:label>
    <link:label id="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxAttributableToReportingEntity_6d1b40d7-70f2-4a00-a505-7b1702a741d3_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_IncomeLossFromDiscontinuedOperationsNetOfTaxAttributableToReportingEntity" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Less: net loss from discontinued operations</link:label>
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    <link:label id="lab_us-gaap_AssumedPremiumsWritten_71fbeac6-e110-45cd-bb65-21d553e2d66d_terseLabel_en-US" xlink:label="lab_us-gaap_AssumedPremiumsWritten" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Assumed</link:label>
    <link:label id="lab_us-gaap_AssumedPremiumsWritten_label_en-US" xlink:label="lab_us-gaap_AssumedPremiumsWritten" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assumed Premiums Written</link:label>
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    <link:label id="lab_us-gaap_UnrealizedGainLossOnInvestmentsTableTextBlock_5af277a0-945e-4766-82fc-addf77802c89_terseLabel_en-US" xlink:label="lab_us-gaap_UnrealizedGainLossOnInvestmentsTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Net Unrealized Losses Included in Accumulated Other Comprehensive Income</link:label>
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    <link:label id="lab_kg_FixedMaturitiesFairValue_c5b26253-8c9e-489c-b3f4-aea2e65ebbc9_verboseLabel_en-US" xlink:label="lab_kg_FixedMaturitiesFairValue" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair value</link:label>
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    <link:label id="lab_kg_FixedMaturitiesFairValue_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Fair Value</link:label>
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    <link:label id="lab_us-gaap_LongTermDebt_7bedc0b7-18ef-4393-b10d-4a53c5daf01b_totalLabel_en-US" xlink:label="lab_us-gaap_LongTermDebt" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Senior notes, net</link:label>
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    <link:label id="lab_dei_LegalEntityAxis_809b26a9-c00b-4dc3-94c6-2abc1e943309_terseLabel_en-US" xlink:label="lab_dei_LegalEntityAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Legal Entity [Axis]</link:label>
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    <link:label id="lab_us-gaap_Assets_eba9427f-9425-46bf-b04d-8d4d4d4af832_totalLabel_en-US" xlink:label="lab_us-gaap_Assets" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total assets</link:label>
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    <link:label id="lab_kg_ThirdPartyAgreementsMember_a5b25dbf-9245-4a86-ac2c-4b67a3ca18e0_terseLabel_en-US" xlink:label="lab_kg_ThirdPartyAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Third Party Agreements</link:label>
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    <link:label id="lab_us-gaap_IncomeTaxPolicyTextBlock_49ffe8f3-007f-4244-a389-d00827003834_terseLabel_en-US" xlink:label="lab_us-gaap_IncomeTaxPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Income Taxes</link:label>
    <link:label id="lab_us-gaap_IncomeTaxPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_IncomeTaxPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Income Tax, Policy [Policy Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_IncomeTaxPolicyTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_IncomeTaxPolicyTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_IncomeTaxPolicyTextBlock" xlink:to="lab_us-gaap_IncomeTaxPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_1fa06fed-4852-4723-8c6b-424889ef98b0_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from operating activities</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Operating Activity, Including Discontinued Operation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:to="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ReinsuranceCollateralByThirdParty_47ff91d6-a0e6-4911-be12-78cac03713a7_terseLabel_en-US" xlink:label="lab_kg_ReinsuranceCollateralByThirdParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collateral by third party</link:label>
    <link:label id="lab_kg_ReinsuranceCollateralByThirdParty_label_en-US" xlink:label="lab_kg_ReinsuranceCollateralByThirdParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance, Collateral By Third Party</link:label>
    <link:label id="lab_kg_ReinsuranceCollateralByThirdParty_documentation_en-US" xlink:label="lab_kg_ReinsuranceCollateralByThirdParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Reinsurance, Collateral By Third Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ReinsuranceCollateralByThirdParty" xlink:href="kg-20250930.xsd#kg_ReinsuranceCollateralByThirdParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ReinsuranceCollateralByThirdParty" xlink:to="lab_kg_ReinsuranceCollateralByThirdParty" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock_e8e030a1-1524-49df-a9ab-e023e5410918_terseLabel_en-US" xlink:label="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Basis of Presentation</link:label>
    <link:label id="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock_label_en-US" xlink:label="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Organization, Consolidation and Presentation of Financial Statements Disclosure and Significant Accounting Policies [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureAndSignificantAccountingPoliciesTextBlock"/>
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    <link:label id="lab_us-gaap_PrivateEquityFundsMember_65f00121-48a5-4eae-8d70-47f079b0ac0a_terseLabel_en-US" xlink:label="lab_us-gaap_PrivateEquityFundsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Private equity funds</link:label>
    <link:label id="lab_us-gaap_PrivateEquityFundsMember_label_en-US" xlink:label="lab_us-gaap_PrivateEquityFundsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Private Equity Funds [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PrivateEquityFundsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PrivateEquityFundsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PrivateEquityFundsMember" xlink:to="lab_us-gaap_PrivateEquityFundsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_VariableInterestEntityLineItems_f402c76c-316c-49fd-928b-fd1ef8647e3f_terseLabel_en-US" xlink:label="lab_us-gaap_VariableInterestEntityLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Variable Interest Entity [Line Items]</link:label>
    <link:label id="lab_us-gaap_VariableInterestEntityLineItems_label_en-US" xlink:label="lab_us-gaap_VariableInterestEntityLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Variable Interest Entity [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_VariableInterestEntityLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_VariableInterestEntityLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_VariableInterestEntityLineItems" xlink:to="lab_us-gaap_VariableInterestEntityLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OtherInvestments_5ad8de16-2b3e-4760-bc44-f678131467c0_terseLabel_en-US" xlink:label="lab_us-gaap_OtherInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other investments</link:label>
    <link:label id="lab_us-gaap_OtherInvestments_054eca22-5060-422d-a84b-d7534089c546_verboseLabel_en-US" xlink:label="lab_us-gaap_OtherInvestments" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Total other investments at fair value</link:label>
    <link:label id="lab_us-gaap_OtherInvestments_label_en-US" xlink:label="lab_us-gaap_OtherInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Other Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OtherInvestments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OtherInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_OtherInvestments" xlink:to="lab_us-gaap_OtherInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityCommonStockSharesOutstanding_4e484b8a-f689-43b8-9c35-b240839e74a5_terseLabel_en-US" xlink:label="lab_dei_EntityCommonStockSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Common Stock, Shares Outstanding</link:label>
    <link:label id="lab_dei_EntityCommonStockSharesOutstanding_label_en-US" xlink:label="lab_dei_EntityCommonStockSharesOutstanding" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Common Stock, Shares Outstanding</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityCommonStockSharesOutstanding" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityCommonStockSharesOutstanding"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityCommonStockSharesOutstanding" xlink:to="lab_dei_EntityCommonStockSharesOutstanding" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems_2b187948-c77e-452f-a592-d428ea84be15_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-sale [Line Items]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems" xlink:to="lab_us-gaap_ScheduleOfAvailableForSaleSecuritiesLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueInputsLevel2Member_6c11b135-e712-4f6a-b0b2-e3618f3dd065_verboseLabel_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel2Member" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Significant Other Observable Inputs (Level 2)</link:label>
    <link:label id="lab_us-gaap_FairValueInputsLevel2Member_label_en-US" xlink:label="lab_us-gaap_FairValueInputsLevel2Member" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Inputs, Level 2 [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueInputsLevel2Member" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueInputsLevel2Member"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueInputsLevel2Member" xlink:to="lab_us-gaap_FairValueInputsLevel2Member" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EquityMethodInvestments_4ccc7db7-792d-4e0f-aae8-3b6ddf64a9a8_verboseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Carrying Value</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestments_label_en-US" xlink:label="lab_us-gaap_EquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquityMethodInvestments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquityMethodInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquityMethodInvestments" xlink:to="lab_us-gaap_EquityMethodInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock_terseLabel_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Not Adopted</link:label>
    <link:label id="lab_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock_label_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Not Adopted [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock" xlink:to="lab_ecd_InsiderTrdPoliciesProcNotAdoptedTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessAcquisitionAcquireeDomain_86e6fcc9-aedb-4d2b-a861-b79e96503db3_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionAcquireeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination [Domain]</link:label>
    <link:label id="lab_us-gaap_BusinessAcquisitionAcquireeDomain_label_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionAcquireeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessAcquisitionAcquireeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessAcquisitionAcquireeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessAcquisitionAcquireeDomain" xlink:to="lab_us-gaap_BusinessAcquisitionAcquireeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyAgreementsMember_aba7ca2f-0b34-48e1-b538-47aeb21aa743_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Agreements</link:label>
    <link:label id="lab_kg_RelatedPartyAgreementsMember_label_en-US" xlink:label="lab_kg_RelatedPartyAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Agreements [Member]</link:label>
    <link:label id="lab_kg_RelatedPartyAgreementsMember_documentation_en-US" xlink:label="lab_kg_RelatedPartyAgreementsMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Agreements [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyAgreementsMember" xlink:href="kg-20250930.xsd#kg_RelatedPartyAgreementsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyAgreementsMember" xlink:to="lab_kg_RelatedPartyAgreementsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EquityMethodInvestmentsMember_72ed7ce6-36e5-4470-8b14-8070ce5895f1_verboseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentsMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Equity method investments with fair value option elected</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestmentsMember_6e16d585-f1dc-43ca-9a54-652a2d3143d9_terseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity method investments measured at fair value option</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestmentsMember_46b73d76-71be-46ca-a63a-623be34adfe3_netLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentsMember" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Total unfunded commitments on equity method investments</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestmentsMember_label_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investments [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquityMethodInvestmentsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquityMethodInvestmentsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquityMethodInvestmentsMember" xlink:to="lab_us-gaap_EquityMethodInvestmentsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DiscontinuedOperationsPolicyTextBlock_2fe0d83a-648f-457e-b52d-e11875ed8b35_terseLabel_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Discontinued Operations</link:label>
    <link:label id="lab_us-gaap_DiscontinuedOperationsPolicyTextBlock_label_en-US" xlink:label="lab_us-gaap_DiscontinuedOperationsPolicyTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Discontinued Operations, Policy [Policy Text Block]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DiscontinuedOperationsPolicyTextBlock" xlink:to="lab_us-gaap_DiscontinuedOperationsPolicyTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PlanNameAxis_82252767-6223-43e8-afde-6256356d4dd0_terseLabel_en-US" xlink:label="lab_us-gaap_PlanNameAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Plan Name [Axis]</link:label>
    <link:label id="lab_us-gaap_PlanNameAxis_label_en-US" xlink:label="lab_us-gaap_PlanNameAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Plan Name [Axis]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PlanNameAxis" xlink:to="lab_us-gaap_PlanNameAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeoMember_terseLabel_en-US" xlink:label="lab_ecd_PeoMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoMember"/>
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    <link:label id="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersIntoLevel3_2aae1ed9-0d36-4ada-a3cf-920e7be9b286_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetTransfersIntoLevel3" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Acquired Level 3 investments</link:label>
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    <link:label id="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_5cdc9885-4505-43ad-84e1-cd2812af6ca1_terseLabel_en-US" xlink:label="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Organization, Consolidation and Presentation of Financial Statements [Abstract]</link:label>
    <link:label id="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract_label_en-US" xlink:label="lab_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Organization, Consolidation and Presentation of Financial Statements [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract"/>
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    <link:label id="lab_kg_DebtInstrumentPriorNoticeRedemptionPeriodStartDate_d9cc2eea-a61b-41de-9cc2-6f1ca10c84d8_terseLabel_en-US" xlink:label="lab_kg_DebtInstrumentPriorNoticeRedemptionPeriodStartDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Prior notice to be given before redemption date</link:label>
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    <link:label id="lab_kg_DebtInstrumentPriorNoticeRedemptionPeriodStartDate_documentation_en-US" xlink:label="lab_kg_DebtInstrumentPriorNoticeRedemptionPeriodStartDate" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Debt Instrument, Prior Notice, Redemption Period, Start Date</link:label>
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    <link:label id="lab_kg_CommonStockNumberOfVotesPerShare_ba664076-af99-4e5c-89ba-f8a250e21583_terseLabel_en-US" xlink:label="lab_kg_CommonStockNumberOfVotesPerShare" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of votes per share</link:label>
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    <link:label id="lab_kg_CommonStockNumberOfVotesPerShare_documentation_en-US" xlink:label="lab_kg_CommonStockNumberOfVotesPerShare" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Common Stock, Number Of Votes Per Share</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CommonStockNumberOfVotesPerShare" xlink:href="kg-20250930.xsd#kg_CommonStockNumberOfVotesPerShare"/>
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    <link:label id="lab_us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock_65e2cf42-eb84-4cbf-8147-b35de2fab131_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfRecognizedIdentifiedAssetsAcquiredAndLiabilitiesAssumedTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Purchase Price of Assets and Liabilities Assumed</link:label>
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    <link:label id="lab_kg_OperatingLeaseEscalatingFeePercentage_cacd1a97-dc0b-48c5-ba15-7c1b1e170558_terseLabel_en-US" xlink:label="lab_kg_OperatingLeaseEscalatingFeePercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Lease payments to escalating fee, percentage</link:label>
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    <link:label id="lab_dei_CityAreaCode_8f9f3f49-e212-42fd-8817-936192577f51_terseLabel_en-US" xlink:label="lab_dei_CityAreaCode" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">City Area Code</link:label>
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    <link:label id="lab_kg_LiabilityForSecuritiesPurchased_7ab5ec5b-87ad-4fa2-81a2-60c4a4dea768_terseLabel_en-US" xlink:label="lab_kg_LiabilityForSecuritiesPurchased" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liability for securities purchased</link:label>
    <link:label id="lab_kg_LiabilityForSecuritiesPurchased_label_en-US" xlink:label="lab_kg_LiabilityForSecuritiesPurchased" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liability For Securities Purchased</link:label>
    <link:label id="lab_kg_LiabilityForSecuritiesPurchased_documentation_en-US" xlink:label="lab_kg_LiabilityForSecuritiesPurchased" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Liability For Securities Purchased</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_LiabilityForSecuritiesPurchased" xlink:href="kg-20250930.xsd#kg_LiabilityForSecuritiesPurchased"/>
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    <link:label id="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable_cecb23c7-16f5-4a68-82f8-54617f0e5554_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Securities, Trading, and Equity Securities, FV-NI [Table]</link:label>
    <link:label id="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable_label_en-US" xlink:label="lab_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Trading, and Equity Securities, FV-NI [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfTradingSecuritiesAndOtherTradingAssetsTable"/>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract_a55f6f9d-9204-48aa-8acd-96a87000923a_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from investing activities:</link:label>
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    <link:label id="lab_us-gaap_GeneralAndAdministrativeExpense_0b66ecb4-7a1f-43cc-8de0-f126ee5cfd31_terseLabel_en-US" xlink:label="lab_us-gaap_GeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">General and administrative expenses</link:label>
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    <link:label id="lab_kg_KestrelGroupLtd.Member_5b941619-cde7-494f-9d98-78356ea47cf9_terseLabel_en-US" xlink:label="lab_kg_KestrelGroupLtd.Member" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Kestrel Group, Ltd.</link:label>
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    <link:label id="lab_ecd_PvpTable_terseLabel_en-US" xlink:label="lab_ecd_PvpTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Pay vs Performance Disclosure</link:label>
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    <link:label id="lab_us-gaap_OperatingLeaseLiabilityStatementOfFinancialPositionExtensibleList_bbff3657-c6e5-41f4-ac00-2dbc687aa895_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseLiabilityStatementOfFinancialPositionExtensibleList" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating Lease, Liability, Statement of Financial Position [Extensible Enumeration]</link:label>
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    <link:label id="lab_us-gaap_InterestReceivable_7103dde6-d440-46c4-a4ba-6eca207035f7_terseLabel_en-US" xlink:label="lab_us-gaap_InterestReceivable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued interest receivable</link:label>
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    <link:label id="lab_us-gaap_RelatedPartyTransactionsAbstract_4635975d-d8e7-497f-814c-486f4bd10f18_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party Transactions [Abstract]</link:label>
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    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredCashConsiderationPaidToShareholders_d10ec484-d4fb-45da-aa7e-fd65023f96dd_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredCashConsiderationPaidToShareholders" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash consideration paid to shareholders at Closing for fractional shares</link:label>
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    <link:label id="lab_us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent_9b7d755c-c476-429d-a13a-8e434e5f7594_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Total equity securities subject to contractual sale restrictions</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent_label_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Securities, FV-NI</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquitySecuritiesFvNiCurrentAndNoncurrent"/>
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    <link:label id="lab_us-gaap_USTreasuryBondSecuritiesMember_d37b8776-661d-43a7-8e17-e7cd45422b7c_terseLabel_en-US" xlink:label="lab_us-gaap_USTreasuryBondSecuritiesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">U.S. treasury bonds</link:label>
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    <link:label id="lab_us-gaap_AvailableForSaleDebtSecuritiesAmortizedCostBasis_1b28e1c6-5289-4a97-a747-404ba998fe63_totalLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleDebtSecuritiesAmortizedCostBasis" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Original or amortized cost</link:label>
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    <link:label id="lab_us-gaap_RelatedPartyDomain_bdfa96eb-e33f-4ffa-bece-66401ed6e4bf_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party [Domain]</link:label>
    <link:label id="lab_us-gaap_RelatedPartyDomain_label_en-US" xlink:label="lab_us-gaap_RelatedPartyDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related and Nonrelated Parties [Domain]</link:label>
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    <link:label id="lab_kg_BarryZyskindMember_7a5f6bb2-3824-4d8e-8514-6a2d1f7831bb_terseLabel_en-US" xlink:label="lab_kg_BarryZyskindMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Barry Zyskind</link:label>
    <link:label id="lab_kg_BarryZyskindMember_label_en-US" xlink:label="lab_kg_BarryZyskindMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Barry Zyskind [Member]</link:label>
    <link:label id="lab_kg_BarryZyskindMember_documentation_en-US" xlink:label="lab_kg_BarryZyskindMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Barry Zyskind</link:label>
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    <link:label id="lab_us-gaap_EquitySecuritiesFvNiGainLoss_4717c688-3dbc-4f6c-bc0c-e3d1c699c904_totalLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiGainLoss" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Equity securities, net</link:label>
    <link:label id="lab_us-gaap_EquitySecuritiesFvNiGainLoss_f65385a0-7137-432f-9f2a-4c32d4378f8c_terseLabel_en-US" xlink:label="lab_us-gaap_EquitySecuritiesFvNiGainLoss" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net gains recognized for equity securities</link:label>
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    <link:label id="lab_us-gaap_FairValueOfFinancialInstrumentsPolicy_4698acc5-0dc2-4f87-b51d-1aad0632861e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueOfFinancialInstrumentsPolicy" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Measurements</link:label>
    <link:label id="lab_us-gaap_FairValueOfFinancialInstrumentsPolicy_label_en-US" xlink:label="lab_us-gaap_FairValueOfFinancialInstrumentsPolicy" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value of Financial Instruments, Policy [Policy Text Block]</link:label>
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    <link:label id="lab_us-gaap_InvestmentsAbstract_307ed909-8f60-43c9-9a18-8a343a298507_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investments:</link:label>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities_73dc6f18-9c2a-492b-86fc-11430c6a17d1_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInInvestingActivities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash provided by investing activities</link:label>
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    <link:label id="lab_kg_UnderwritingIncomeLossNet_documentation_en-US" xlink:label="lab_kg_UnderwritingIncomeLossNet" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Net premiums earned less: net losses and loss expenses, acquisition costs and general and administrative expenses directly related to underwriting activities.</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_UnderwritingIncomeLossNet" xlink:href="kg-20250930.xsd#kg_UnderwritingIncomeLossNet"/>
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    <link:label id="lab_ecd_Additional402vDisclosureTextBlock_label_en-US" xlink:label="lab_ecd_Additional402vDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Additional 402(v) Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_Additional402vDisclosureTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_Additional402vDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_Additional402vDisclosureTextBlock" xlink:to="lab_ecd_Additional402vDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember_22fd339b-9a8b-4886-b82d-aa0316992b9a_terseLabel_en-US" xlink:label="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement and Arrangement Other than Collaborative [Domain]</link:label>
    <link:label id="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember_label_en-US" xlink:label="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement and Arrangement Other than Collaborative [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ArrangementsAndNonarrangementTransactionsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:to="lab_us-gaap_ArrangementsAndNonarrangementTransactionsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FinancingReceivableAccruedInterestWriteoff_2b42fa0d-5cdf-4553-a7ed-d4baa5c95037_terseLabel_en-US" xlink:label="lab_us-gaap_FinancingReceivableAccruedInterestWriteoff" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Write off on the accrued interest receivable</link:label>
    <link:label id="lab_us-gaap_FinancingReceivableAccruedInterestWriteoff_label_en-US" xlink:label="lab_us-gaap_FinancingReceivableAccruedInterestWriteoff" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Financing Receivable, Accrued Interest, Writeoff</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FinancingReceivableAccruedInterestWriteoff" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FinancingReceivableAccruedInterestWriteoff"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FinancingReceivableAccruedInterestWriteoff" xlink:to="lab_us-gaap_FinancingReceivableAccruedInterestWriteoff" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PaymentsForProceedsFromOtherInvestingActivities_33a4f27e-d6d3-4496-b242-a9ec28588f44_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_PaymentsForProceedsFromOtherInvestingActivities" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">Others, net</link:label>
    <link:label id="lab_us-gaap_PaymentsForProceedsFromOtherInvestingActivities_label_en-US" xlink:label="lab_us-gaap_PaymentsForProceedsFromOtherInvestingActivities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Payment for (Proceeds from) Other Investing Activity</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PaymentsForProceedsFromOtherInvestingActivities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PaymentsForProceedsFromOtherInvestingActivities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_PaymentsForProceedsFromOtherInvestingActivities" xlink:to="lab_us-gaap_PaymentsForProceedsFromOtherInvestingActivities" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill_4bd35e73-4016-4970-ab77-cdc2c4ea9ce6_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intangible assets</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill_02939061-5156-4fe9-8f73-676f66b8bb40_totalLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total Intangible Assets at acquisition</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Recognized Asset Acquired, Identifiable Intangible Asset, Excluding Goodwill</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:to="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedIntangibleAssetsOtherThanGoodwill" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_4a3bd15a-b848-4949-bb64-146bc5ae0f52_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Recognized Asset Acquired, Cash and Cash Equivalent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationRecognizedIdentifiableAssetsAcquiredAndLiabilitiesAssumedCashAndEquivalents"/>
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    <link:label id="lab_us-gaap_AssetsAbstract_0e2dd004-c98e-47bb-89c0-1de7df9da8df_terseLabel_en-US" xlink:label="lab_us-gaap_AssetsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">ASSETS</link:label>
    <link:label id="lab_us-gaap_AssetsAbstract_label_en-US" xlink:label="lab_us-gaap_AssetsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assets [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AssetsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AssetsAbstract" xlink:to="lab_us-gaap_AssetsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember_850903b0-5c61-424e-9eff-64ff832bb137_terseLabel_en-US" xlink:label="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement, Minimum Excess Funding Requirement</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember_label_en-US" xlink:label="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement, Minimum Excess Funding Requirement [Member]</link:label>
    <link:label id="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember_documentation_en-US" xlink:label="lab_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Post-Termination Endorsement, Minimum Excess Funding Requirement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember" xlink:href="kg-20250930.xsd#kg_PostTerminationEndorsementMinimumExcessFundingRequirementMember"/>
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    <link:label id="lab_ecd_UndrlygSecurityMktPriceChngPct_terseLabel_en-US" xlink:label="lab_ecd_UndrlygSecurityMktPriceChngPct" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Underlying Security Market Price Change</link:label>
    <link:label id="lab_ecd_UndrlygSecurityMktPriceChngPct_label_en-US" xlink:label="lab_ecd_UndrlygSecurityMktPriceChngPct" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Underlying Security Market Price Change, Percent</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_UndrlygSecurityMktPriceChngPct" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_UndrlygSecurityMktPriceChngPct"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_UndrlygSecurityMktPriceChngPct" xlink:to="lab_ecd_UndrlygSecurityMktPriceChngPct" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentAxis_afe5b805-9a3c-4434-8178-fb5402bfd598_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt Instrument [Axis]</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentAxis_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentAxis" xlink:to="lab_us-gaap_DebtInstrumentAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LoansAndLeasesReceivableRelatedParties_c954d865-39c2-4c53-9341-c3117174d136_terseLabel_en-US" xlink:label="lab_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net loan receivable from related party</link:label>
    <link:label id="lab_us-gaap_LoansAndLeasesReceivableRelatedParties_dc52bac9-1c41-49d2-8e40-3d5f6962cce7_verboseLabel_en-US" xlink:label="lab_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Loan to related party</link:label>
    <link:label id="lab_us-gaap_LoansAndLeasesReceivableRelatedParties_label_en-US" xlink:label="lab_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Loans and Leases Receivable, Related Parties</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LoansAndLeasesReceivableRelatedParties"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:to="lab_us-gaap_LoansAndLeasesReceivableRelatedParties" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss_b921b702-ed57-41f7-9cec-d5983b39b5fe_negatedLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">12 Months or More</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss_label_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, 12 Months or Longer, Accumulated Loss</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, 12 Months or Longer, Accumulated Loss</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:to="lab_kg_FixedMaturitiesContinuousUnrealizedLossPosition12MonthsorLongerAccumulatedLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueDisclosuresTextBlock_a75b80ac-5816-4d9e-8e0f-f31c239ab29f_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueDisclosuresTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value of Financial Instruments</link:label>
    <link:label id="lab_us-gaap_FairValueDisclosuresTextBlock_label_en-US" xlink:label="lab_us-gaap_FairValueDisclosuresTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Disclosures [Text Block]</link:label>
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    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueDisclosuresTextBlock" xlink:to="lab_us-gaap_FairValueDisclosuresTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessAcquisitionLineItems_006e505f-cc71-493e-8008-dcb440589698_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination [Line Items]</link:label>
    <link:label id="lab_us-gaap_BusinessAcquisitionLineItems_label_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessAcquisitionLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessAcquisitionLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessAcquisitionLineItems" xlink:to="lab_us-gaap_BusinessAcquisitionLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost_255423bb-cbed-42e5-a62c-41111c337fb2_terseLabel_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative arrangement, reserves expected to be commuted, exit cost</link:label>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost_label_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Reserves Expected to be Commuted, Exit Cost</link:label>
    <link:label id="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost_documentation_en-US" xlink:label="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement, Reserves Expected to be Commuted, Exit Cost</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:href="kg-20250930.xsd#kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:to="lab_kg_CollaborativeArrangementReservesExpectedToBeCommutedExitCost" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_df285f35-b3b3-42b6-9e0b-928c946f0980_verboseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Stated interest rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_d5851d57-3155-4352-b4fd-e2c8efbae50e_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Coupon rate</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Interest Rate, Stated Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentInterestRateStatedPercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:to="lab_us-gaap_DebtInstrumentInterestRateStatedPercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis_f64c76c1-1131-485c-80e6-fbba65ea1e48_terseLabel_en-US" xlink:label="lab_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment, Name [Axis]</link:label>
    <link:label id="lab_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis_label_en-US" xlink:label="lab_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment, Name [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis" xlink:href="https://xbrl.fasb.org/srt/2025/elts/srt-2025.xsd#srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis" xlink:to="lab_srt_ScheduleOfEquityMethodInvestmentEquityMethodInvesteeNameAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock_c6ed2fba-f9e3-4021-9185-186c50e9e112_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Net Premiums by Major Line of Business</link:label>
    <link:label id="lab_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Revenue from External Customers by Products and Services [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock" xlink:to="lab_us-gaap_ScheduleOfEntityWideInformationRevenueFromExternalCustomersByProductsAndServicesTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue_9398e980-e135-4e3a-aba5-1093fa3bb283_terseLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Less than 12 Months</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue_label_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Less than Twelve Months, Fair Value</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Less than Twelve Months, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:to="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthanTwelveMonthsFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EntitiesOwnedByRelatedPartiesAxis_0560e6f7-31cb-4bf1-a43a-6aa77f45e7cc_terseLabel_en-US" xlink:label="lab_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entities Owned by Related Parties [Axis]</link:label>
    <link:label id="lab_kg_EntitiesOwnedByRelatedPartiesAxis_label_en-US" xlink:label="lab_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entities Owned by Related Parties [Axis]</link:label>
    <link:label id="lab_kg_EntitiesOwnedByRelatedPartiesAxis_documentation_en-US" xlink:label="lab_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Entities Owned by Related Parties [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:href="kg-20250930.xsd#kg_EntitiesOwnedByRelatedPartiesAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:to="lab_kg_EntitiesOwnedByRelatedPartiesAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DepreciationAmortizationAndAccretionNet_d1ad751e-dc9f-4c10-9377-4e94392a2302_terseLabel_en-US" xlink:label="lab_us-gaap_DepreciationAmortizationAndAccretionNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Other non-cash expenses (credit losses, depreciation, amortization, share-based compensation, leases)</link:label>
    <link:label id="lab_us-gaap_DepreciationAmortizationAndAccretionNet_label_en-US" xlink:label="lab_us-gaap_DepreciationAmortizationAndAccretionNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Depreciation, Amortization and Accretion, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DepreciationAmortizationAndAccretionNet" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DepreciationAmortizationAndAccretionNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DepreciationAmortizationAndAccretionNet" xlink:to="lab_us-gaap_DepreciationAmortizationAndAccretionNet" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_EntityRegistrantName_ee8e6ed0-8c5e-4283-80b1-0575426b53de_terseLabel_en-US" xlink:label="lab_dei_EntityRegistrantName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Entity Registrant Name</link:label>
    <link:label id="lab_dei_EntityRegistrantName_label_en-US" xlink:label="lab_dei_EntityRegistrantName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Entity Registrant Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_EntityRegistrantName" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_EntityRegistrantName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_dei_EntityRegistrantName" xlink:to="lab_dei_EntityRegistrantName" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock_terseLabel_en-US" xlink:label="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Stock Price or TSR Estimation Method</link:label>
    <link:label id="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock_label_en-US" xlink:label="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Stock Price or TSR Estimation Method [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_StkPrcOrTsrEstimationMethodTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:to="lab_ecd_StkPrcOrTsrEstimationMethodTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_BusinessAcquisitionOptionTerm_6aec3dbb-eb4c-4055-a99e-494c6e389eff_terseLabel_en-US" xlink:label="lab_kg_BusinessAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Acquire option term</link:label>
    <link:label id="lab_kg_BusinessAcquisitionOptionTerm_label_en-US" xlink:label="lab_kg_BusinessAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Acquisition, Option Term</link:label>
    <link:label id="lab_kg_BusinessAcquisitionOptionTerm_documentation_en-US" xlink:label="lab_kg_BusinessAcquisitionOptionTerm" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Acquisition, Option Term</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessAcquisitionOptionTerm" xlink:href="kg-20250930.xsd#kg_BusinessAcquisitionOptionTerm"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessAcquisitionOptionTerm" xlink:to="lab_kg_BusinessAcquisitionOptionTerm" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AmTrustLegacySegmentMember_eb9b8e46-8a99-4735-a897-d3f26ea5dab1_verboseLabel_en-US" xlink:label="lab_kg_AmTrustLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">AmTrust Legacy Reinsurance</link:label>
    <link:label id="lab_kg_AmTrustLegacySegmentMember_label_en-US" xlink:label="lab_kg_AmTrustLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AmTrust Legacy Segment [Member]</link:label>
    <link:label id="lab_kg_AmTrustLegacySegmentMember_documentation_en-US" xlink:label="lab_kg_AmTrustLegacySegmentMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">AmTrust Legacy Segment</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AmTrustLegacySegmentMember" xlink:href="kg-20250930.xsd#kg_AmTrustLegacySegmentMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_AmTrustLegacySegmentMember" xlink:to="lab_kg_AmTrustLegacySegmentMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation_41b6fbff-6ae4-4819-9663-6683e6f1ff22_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liabilities held for sale</link:label>
    <link:label id="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation_687c8a18-306e-44c5-ba6a-04967d4a6ec6_totalLabel_en-US" xlink:label="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Liabilities held for sale</link:label>
    <link:label id="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation_label_en-US" xlink:label="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:to="lab_us-gaap_LiabilitiesOfDisposalGroupIncludingDiscontinuedOperation" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies_04896b02-8845-4a1c-bd1b-77f2147daeb1_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Future premiums and losses related to policies, percentage</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Future Premiums and Losses Related to Policies</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Future Premiums and Losses Related to Policies</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:to="lab_kg_RelatedPartyTransactionFuturePremiumsandLossesRelatedtoPolicies" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_AssetsFairValueDisclosure_e6da45a7-48a9-4564-9941-8fd99bb60704_totalLabel_en-US" xlink:label="lab_us-gaap_AssetsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total investments</link:label>
    <link:label id="lab_us-gaap_AssetsFairValueDisclosure_cee0f8fb-914e-4f2b-aed2-6f6627e230d3_terseLabel_en-US" xlink:label="lab_us-gaap_AssetsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Financial instruments, fair value</link:label>
    <link:label id="lab_us-gaap_AssetsFairValueDisclosure_bdcb75a0-1dd9-429d-920b-fbb591baf152_verboseLabel_en-US" xlink:label="lab_us-gaap_AssetsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Fair Value</link:label>
    <link:label id="lab_us-gaap_AssetsFairValueDisclosure_label_en-US" xlink:label="lab_us-gaap_AssetsFairValueDisclosure" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Assets, Fair Value Disclosure</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AssetsFairValueDisclosure" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AssetsFairValueDisclosure"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AssetsFairValueDisclosure" xlink:to="lab_us-gaap_AssetsFairValueDisclosure" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching_0a5ea70f-4b9a-4a46-b686-ece6369f8e50_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percent basis currency equivalent of maximum limit of liability attaching</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent Basis Currency Equivalent of Maximum Limit of Liability Attaching</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Percent Basis Currency Equivalent of Maximum Limit of Liability Attaching</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:to="lab_kg_RelatedPartyTransactionPercentBasisCurrencyEquivalentOfMaximumLimitOfLiabilityAttaching" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock_64943559-65cc-43a3-a82c-c6a434a5a039_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Allowance for Expected Credit Losses on Reinsurance Recoverable</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Recoverable, Allowance for Credit Loss [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock" xlink:to="lab_us-gaap_ReinsuranceRecoverableAllowanceForCreditLossTableTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_471a39c9-6ff6-4978-9597-b189cd889396_terseLabel_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Related Party [Axis]</link:label>
    <link:label id="lab_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis_label_en-US" xlink:label="lab_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related and Nonrelated Parties [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RelatedPartyTransactionsByRelatedPartyAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:to="lab_us-gaap_RelatedPartyTransactionsByRelatedPartyAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReinsuranceRetentionPolicyLineItems_0b68f3c7-bff7-4b4d-a6ad-412169693e49_terseLabel_en-US" xlink:label="lab_us-gaap_ReinsuranceRetentionPolicyLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reinsurance Retention Policy [Line Items]</link:label>
    <link:label id="lab_us-gaap_ReinsuranceRetentionPolicyLineItems_label_en-US" xlink:label="lab_us-gaap_ReinsuranceRetentionPolicyLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reinsurance Retention [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReinsuranceRetentionPolicyLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReinsuranceRetentionPolicyLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReinsuranceRetentionPolicyLineItems" xlink:to="lab_us-gaap_ReinsuranceRetentionPolicyLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations_ee330972-c86e-4fa0-8566-5f6fb20b0867_totalLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net cash used in continuing operations</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Operating Activity, Continuing Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations" xlink:to="lab_us-gaap_NetCashProvidedByUsedInOperatingActivitiesContinuingOperations" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_TypeOfArrangementAxis_50df7cc0-add5-4c31-a598-393821b43cce_terseLabel_en-US" xlink:label="lab_us-gaap_TypeOfArrangementAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement and Arrangement Other than Collaborative [Axis]</link:label>
    <link:label id="lab_us-gaap_TypeOfArrangementAxis_label_en-US" xlink:label="lab_us-gaap_TypeOfArrangementAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Collaborative Arrangement and Arrangement Other than Collaborative [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TypeOfArrangementAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TypeOfArrangementAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TypeOfArrangementAxis" xlink:to="lab_us-gaap_TypeOfArrangementAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_RestatementDateAxis_terseLabel_en-US" xlink:label="lab_ecd_RestatementDateAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restatement Determination Date:</link:label>
    <link:label id="lab_ecd_RestatementDateAxis_label_en-US" xlink:label="lab_ecd_RestatementDateAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restatement Determination Date [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_RestatementDateAxis" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_RestatementDateAxis"/>
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    <link:label id="lab_kg_ControllingInterestOwnershipPercentage_367af8f0-8cf3-4524-bb11-72266cf8121b_terseLabel_en-US" xlink:label="lab_kg_ControllingInterestOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Controlling interest, ownership percentage</link:label>
    <link:label id="lab_kg_ControllingInterestOwnershipPercentage_label_en-US" xlink:label="lab_kg_ControllingInterestOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Controlling Interest, Ownership Percentage</link:label>
    <link:label id="lab_kg_ControllingInterestOwnershipPercentage_documentation_en-US" xlink:label="lab_kg_ControllingInterestOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Controlling Interest, Ownership Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ControllingInterestOwnershipPercentage" xlink:href="kg-20250930.xsd#kg_ControllingInterestOwnershipPercentage"/>
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    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement_5d314de4-cfc7-4a93-9684-9ba1e89f9694_terseLabel_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Intercompany settlement</link:label>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement_label_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Consideration Transferred, Intercompany Settlement</link:label>
    <link:label id="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement_documentation_en-US" xlink:label="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Business Combination, Consideration Transferred, Intercompany Settlement</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:href="kg-20250930.xsd#kg_BusinessCombinationConsiderationTransferredIntercompanySettlement"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:to="lab_kg_BusinessCombinationConsiderationTransferredIntercompanySettlement" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AmTrustNorthAmericaIncMember_365c3701-0d3b-4cf9-aa37-665bc661afb3_terseLabel_en-US" xlink:label="lab_kg_AmTrustNorthAmericaIncMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">AmTrust North America Inc</link:label>
    <link:label id="lab_kg_AmTrustNorthAmericaIncMember_label_en-US" xlink:label="lab_kg_AmTrustNorthAmericaIncMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AmTrust North America Inc [Member]</link:label>
    <link:label id="lab_kg_AmTrustNorthAmericaIncMember_documentation_en-US" xlink:label="lab_kg_AmTrustNorthAmericaIncMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">AmTrust North America Inc</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AmTrustNorthAmericaIncMember" xlink:href="kg-20250930.xsd#kg_AmTrustNorthAmericaIncMember"/>
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    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable_b7c04e63-ff4c-49dc-820f-897745d8b799_terseLabel_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accumulated Other Comprehensive Income (Loss) [Table]</link:label>
    <link:label id="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable_label_en-US" xlink:label="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Accumulated Other Comprehensive Income (Loss) [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AccumulatedOtherComprehensiveIncomeLossTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable" xlink:to="lab_us-gaap_AccumulatedOtherComprehensiveIncomeLossTable" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty_3bdeaa2b-12f7-477e-8219-1d315257d546_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued interest on collateral held by related party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty_label_en-US" xlink:label="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Accrued Interest on Collateral Held by Related Party</link:label>
    <link:label id="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty_documentation_en-US" xlink:label="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Related Party Transaction, Accrued Interest on Collateral Held by Related Party</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:href="kg-20250930.xsd#kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:to="lab_kg_RelatedPartyTransactionAccruedInterestOnCollateralHeldByRelatedParty" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InterestExpenseDebt_6cdd8421-03b3-4f2b-90d9-84ed17a31573_terseLabel_en-US" xlink:label="lab_us-gaap_InterestExpenseDebt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Interest and amortization expenses</link:label>
    <link:label id="lab_us-gaap_InterestExpenseDebt_label_en-US" xlink:label="lab_us-gaap_InterestExpenseDebt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Interest Expense, Debt</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InterestExpenseDebt" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InterestExpenseDebt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InterestExpenseDebt" xlink:to="lab_us-gaap_InterestExpenseDebt" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_RestrictedInvestments_2da7e279-a635-41f7-937b-c142c581859c_terseLabel_en-US" xlink:label="lab_us-gaap_RestrictedInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted investments</link:label>
    <link:label id="lab_us-gaap_RestrictedInvestments_label_en-US" xlink:label="lab_us-gaap_RestrictedInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_RestrictedInvestments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_RestrictedInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_RestrictedInvestments" xlink:to="lab_us-gaap_RestrictedInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_SegmentReportingRevenueReconcilingItemLineItems_68e97319-28cc-4501-a470-bd7c72135098_terseLabel_en-US" xlink:label="lab_us-gaap_SegmentReportingRevenueReconcilingItemLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Segment Reporting, Revenue Reconciling Item [Line Items]</link:label>
    <link:label id="lab_us-gaap_SegmentReportingRevenueReconcilingItemLineItems_label_en-US" xlink:label="lab_us-gaap_SegmentReportingRevenueReconcilingItemLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Segment Reporting, Revenue Reconciling Item [Line Items]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SegmentReportingRevenueReconcilingItemLineItems" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SegmentReportingRevenueReconcilingItemLineItems"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_SegmentReportingRevenueReconcilingItemLineItems" xlink:to="lab_us-gaap_SegmentReportingRevenueReconcilingItemLineItems" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrIndName_terseLabel_en-US" xlink:label="lab_ecd_TrdArrIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_TrdArrIndName_label_en-US" xlink:label="lab_ecd_TrdArrIndName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement, Individual Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrIndName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrIndName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_TrdArrIndName" xlink:to="lab_ecd_TrdArrIndName" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable_2de2401f-c123-45b7-8666-60833e69c26a_terseLabel_en-US" xlink:label="lab_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reconciliation of Revenue from Segments to Consolidated [Table]</link:label>
    <link:label id="lab_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable_label_en-US" xlink:label="lab_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reconciliation of Revenue from Segments to Consolidated [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable" xlink:to="lab_us-gaap_ReconciliationOfRevenueFromSegmentsToConsolidatedTable" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss_124aae20-f3c5-4475-bfce-514a0b5be439_negatedLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Less than 12 Months</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss_label_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Less than 12 Months, Accumulated Loss</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss" xlink:to="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionLessthan12MonthsAccumulatedLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_LeahKarfunkelMember_0af177d6-54ab-477f-9157-ccd644723b99_terseLabel_en-US" xlink:label="lab_kg_LeahKarfunkelMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Leah Karfunkel</link:label>
    <link:label id="lab_kg_LeahKarfunkelMember_label_en-US" xlink:label="lab_kg_LeahKarfunkelMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Leah Karfunkel [Member]</link:label>
    <link:label id="lab_kg_LeahKarfunkelMember_documentation_en-US" xlink:label="lab_kg_LeahKarfunkelMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Leah Karfunkel</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_LeahKarfunkelMember" xlink:href="kg-20250930.xsd#kg_LeahKarfunkelMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_LeahKarfunkelMember" xlink:to="lab_kg_LeahKarfunkelMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtInstrumentFaceAmount_54415aea-a2a5-4458-87fc-dd5baf7a1870_terseLabel_en-US" xlink:label="lab_us-gaap_DebtInstrumentFaceAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Debt instrument, principal amount</link:label>
    <link:label id="lab_us-gaap_DebtInstrumentFaceAmount_label_en-US" xlink:label="lab_us-gaap_DebtInstrumentFaceAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Instrument, Face Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtInstrumentFaceAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtInstrumentFaceAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtInstrumentFaceAmount" xlink:to="lab_us-gaap_DebtInstrumentFaceAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract_8f27636d-345f-4f42-a138-959c28040440_terseLabel_en-US" xlink:label="lab_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement of Comprehensive Income [Abstract]</link:label>
    <link:label id="lab_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract_label_en-US" xlink:label="lab_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Statement of Comprehensive Income [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:to="lab_us-gaap_StatementOfIncomeAndComprehensiveIncomeAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable_24d008af-8dcd-4378-8143-f7745ca5bdfc_totalLabel_en-US" xlink:label="lab_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net equity portion of consideration transferred at Closing</link:label>
    <link:label id="lab_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable_label_en-US" xlink:label="lab_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination, Consideration Transferred, Equity Interest</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:to="lab_us-gaap_BusinessCombinationConsiderationTransferredEquityInterestsIssuedAndIssuable" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PreferredStockMember_e3766c7a-7b4c-43b3-aff4-e3053fa941ae_terseLabel_en-US" xlink:label="lab_us-gaap_PreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Privately held preferred stocks</link:label>
    <link:label id="lab_us-gaap_PreferredStockMember_e4c21ea1-3de2-4105-84d6-142c51648af8_netLabel_en-US" xlink:label="lab_us-gaap_PreferredStockMember" xlink:role="http://www.xbrl.org/2009/role/netLabel" xlink:type="resource" xml:lang="en-US">Preferred Shares</link:label>
    <link:label id="lab_us-gaap_PreferredStockMember_label_en-US" xlink:label="lab_us-gaap_PreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Preferred Stock [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_PreferredStockMember" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_PreferredStockMember"/>
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    <link:label id="lab_ecd_AwardsCloseToMnpiDiscIndName_terseLabel_en-US" xlink:label="lab_ecd_AwardsCloseToMnpiDiscIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
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    <link:label id="lab_kg_NotesReceivableRelatedPartyMember_documentation_en-US" xlink:label="lab_kg_NotesReceivableRelatedPartyMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Notes Receivable, Related Party [Member]</link:label>
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    <link:label id="lab_ecd_AllExecutiveCategoriesMember_terseLabel_en-US" xlink:label="lab_ecd_AllExecutiveCategoriesMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Executive Categories</link:label>
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    <link:label id="lab_us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersCurrentYearClaimsAndClaimsAdjustmentExpense_33367f9f-140c-4b70-b426-be63326d8a41_verboseLabel_en-US" xlink:label="lab_us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersCurrentYearClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Net incurred losses related to current year</link:label>
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    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersCurrentYearClaimsAndClaimsAdjustmentExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_SupplementalInformationForPropertyCasualtyInsuranceUnderwritersCurrentYearClaimsAndClaimsAdjustmentExpense"/>
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    <link:label id="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense_b697c335-5618-4355-9b6f-1bef0a079457_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commission and other acquisition expenses</link:label>
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    <link:label id="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense_7f847efb-0ae2-46bd-984f-93ce43fb9d2a_negatedLabel_en-US" xlink:label="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Commission and other acquisition expenses</link:label>
    <link:label id="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense_label_en-US" xlink:label="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Deferred Policy Acquisition Costs, Amortization Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:to="lab_us-gaap_DeferredPolicyAcquisitionCostAmortizationExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CommonStockValue_1a661f24-0487-46ec-a467-525a90db8b5a_terseLabel_en-US" xlink:label="lab_us-gaap_CommonStockValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares ($0.01 par value; 2025: 9,979,477 and 2024 - 2,749,996 shares issued; 2025: 7,741,943 and 2024 - 2,749,996 shares outstanding)</link:label>
    <link:label id="lab_us-gaap_CommonStockValue_label_en-US" xlink:label="lab_us-gaap_CommonStockValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Common Stock, Value, Issued</link:label>
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    <link:label id="lab_us-gaap_ShortTermInvestments_aa34b987-42c3-4727-9341-42735475403a_terseLabel_en-US" xlink:label="lab_us-gaap_ShortTermInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Short-term investments</link:label>
    <link:label id="lab_us-gaap_ShortTermInvestments_label_en-US" xlink:label="lab_us-gaap_ShortTermInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Short-Term Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShortTermInvestments" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShortTermInvestments"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShortTermInvestments" xlink:to="lab_us-gaap_ShortTermInvestments" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_PremiumsEarnedNet_932fde7d-7307-4785-9c95-c882094b6e10_totalLabel_en-US" xlink:label="lab_us-gaap_PremiumsEarnedNet" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Net premiums earned</link:label>
    <link:label id="lab_us-gaap_PremiumsEarnedNet_12fe2a31-ba90-4cd0-85f1-4577ffb470d2_verboseLabel_en-US" xlink:label="lab_us-gaap_PremiumsEarnedNet" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Net premiums earned</link:label>
    <link:label id="lab_us-gaap_PremiumsEarnedNet_b1e96f49-0772-4c6b-ac10-d710e6888b37_terseLabel_en-US" xlink:label="lab_us-gaap_PremiumsEarnedNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net premiums earned</link:label>
    <link:label id="lab_us-gaap_PremiumsEarnedNet_label_en-US" xlink:label="lab_us-gaap_PremiumsEarnedNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Premiums Earned, Net</link:label>
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    <link:label id="lab_kg_MarketableSecuritiesRealizedLoss_8f710822-3691-4e05-aed8-0a15ec5168c5_negatedLabel_en-US" xlink:label="lab_kg_MarketableSecuritiesRealizedLoss" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Other investments, gross losses</link:label>
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    <link:label id="lab_kg_MarketableSecuritiesRealizedLoss_documentation_en-US" xlink:label="lab_kg_MarketableSecuritiesRealizedLoss" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Marketable Securities, Realized Loss</link:label>
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    <link:label id="lab_us-gaap_CommitmentsAndContingencies_1aba48ec-8c00-4dcb-98c4-4b093cbf06d7_terseLabel_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingencies" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies</link:label>
    <link:label id="lab_us-gaap_CommitmentsAndContingencies_label_en-US" xlink:label="lab_us-gaap_CommitmentsAndContingencies" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Commitments and Contingencies</link:label>
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    <link:label id="lab_us-gaap_IncreaseDecreaseInOperatingLiabilitiesAbstract_7c6016f8-0166-4157-8851-5b1bd071ab58_terseLabel_en-US" xlink:label="lab_us-gaap_IncreaseDecreaseInOperatingLiabilitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Changes in liabilities &#8211; increase (decrease):</link:label>
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    <link:label id="lab_us-gaap_CashAcquiredFromAcquisition_ae94b48a-e008-4070-b726-a84928648fc8_terseLabel_en-US" xlink:label="lab_us-gaap_CashAcquiredFromAcquisition" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash acquired from purchase of business</link:label>
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    <link:label id="lab_us-gaap_FairValueMeasuredAtNetAssetValuePerShareMember_76f0c5a4-0f62-4e32-b600-875a90ea5570_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueMeasuredAtNetAssetValuePerShareMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Based on NAV Practical Expedient</link:label>
    <link:label id="lab_us-gaap_FairValueMeasuredAtNetAssetValuePerShareMember_label_en-US" xlink:label="lab_us-gaap_FairValueMeasuredAtNetAssetValuePerShareMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Measured at Net Asset Value Per Share [Member]</link:label>
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    <link:label id="lab_kg_EquityMethodInvestmentsDistributionsNetPresentValue_4edb607e-10dc-4283-b785-e1fa4e11035e_terseLabel_en-US" xlink:label="lab_kg_EquityMethodInvestmentsDistributionsNetPresentValue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net present value of total distributions</link:label>
    <link:label id="lab_kg_EquityMethodInvestmentsDistributionsNetPresentValue_label_en-US" xlink:label="lab_kg_EquityMethodInvestmentsDistributionsNetPresentValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investments, Distributions, Net Present Value</link:label>
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    <link:label id="lab_us-gaap_BusinessAcquisitionAxis_21588471-6b7a-418f-a2c1-4b4cd4a42507_terseLabel_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Business Combination [Axis]</link:label>
    <link:label id="lab_us-gaap_BusinessAcquisitionAxis_label_en-US" xlink:label="lab_us-gaap_BusinessAcquisitionAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_BusinessAcquisitionAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_BusinessAcquisitionAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_BusinessAcquisitionAxis" xlink:to="lab_us-gaap_BusinessAcquisitionAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue_d3813669-cce2-4a17-8070-d8aae35b9966_totalLabel_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue_label_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Fair Value</link:label>
    <link:label id="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue_documentation_en-US" xlink:label="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Fixed Maturities, Continuous Unrealized Loss Position, Fair Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:href="kg-20250930.xsd#kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:to="lab_kg_FixedMaturitiesContinuousUnrealizedLossPositionFairValue" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember_626bc91d-ce53-4433-9451-4ca578499839_terseLabel_en-US" xlink:label="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">AmTrust European Hospital Liability Quota Share Agreement</link:label>
    <link:label id="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember_label_en-US" xlink:label="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">AmTrust European Hospital Liability Quota Share Agreement [Member]</link:label>
    <link:label id="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember_documentation_en-US" xlink:label="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">AmTrust European Hospital Liability Quota Share Agreement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:href="kg-20250930.xsd#kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:to="lab_kg_AmtrustEuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeDomain_e241e2fe-e1c8-434a-9e6d-7f6a6cd941e2_terseLabel_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Domain]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeDomain_label_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Domain]</link:label>
    <link:label id="lab_kg_RestrictedCashAndInvestmentsTypeDomain_documentation_en-US" xlink:label="lab_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Restricted Cash and Investments, Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:href="kg-20250930.xsd#kg_RestrictedCashAndInvestmentsTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:to="lab_kg_RestrictedCashAndInvestmentsTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets_dd89f4ab-a444-4a2c-9a53-7746ca8ac27f_terseLabel_en-US" xlink:label="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">As a percentage of total assets</link:label>
    <link:label id="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets_label_en-US" xlink:label="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Percent of Assets Measured at Fair Value to Total Assets</link:label>
    <link:label id="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets_documentation_en-US" xlink:label="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Percent of Assets Measured at Fair Value to Total Assets</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:href="kg-20250930.xsd#kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:to="lab_kg_PercentOfAssetsMeasuredAtFairValueToTotalAssets" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_InvestmentIncomeTextBlock_aa78ce33-6367-4ed7-a0e4-c74d69e14ff1_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentIncomeTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Net Investment Income</link:label>
    <link:label id="lab_us-gaap_InvestmentIncomeTextBlock_label_en-US" xlink:label="lab_us-gaap_InvestmentIncomeTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Income [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentIncomeTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentIncomeTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentIncomeTextBlock" xlink:to="lab_us-gaap_InvestmentIncomeTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage_ea15ff83-48a5-4b6a-9917-525051ce5b8e_verboseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Ownership percentage</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage_ef292b17-92a3-438d-ba78-be7143fb1bd4_terseLabel_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Beneficial Ownership</link:label>
    <link:label id="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage_label_en-US" xlink:label="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investment, Ownership Percentage</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_EquityMethodInvestmentOwnershipPercentage"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:to="lab_us-gaap_EquityMethodInvestmentOwnershipPercentage" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_89a74d74-fac3-4051-a8de-e78677bd0988_terseLabel_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Numerator:</link:label>
    <link:label id="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract_label_en-US" xlink:label="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Net Income (Loss) Available to Common Stockholders, Basic [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:to="lab_us-gaap_NetIncomeLossAvailableToCommonStockholdersBasicAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount_32a7565f-322d-4262-84d8-7cfebd8a5286_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reserve for losses incurred but not reported ("IBNR")</link:label>
    <link:label id="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount_label_en-US" xlink:label="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liability for Unpaid Claims and Claims Adjustment Expense, Incurred but Not Reported (IBNR) Claims, Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount" xlink:to="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseIncurredButNotReportedIBNRClaimsAmount" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ReconciliationtoNetIncomeLossAbstract_6ef08970-8951-4ed7-afdd-a20822dae340_terseLabel_en-US" xlink:label="lab_kg_ReconciliationtoNetIncomeLossAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reconciliation to net income (losses) from continuing operations</link:label>
    <link:label id="lab_kg_ReconciliationtoNetIncomeLossAbstract_label_en-US" xlink:label="lab_kg_ReconciliationtoNetIncomeLossAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Reconciliation to Net Income (Loss) [Abstract]</link:label>
    <link:label id="lab_kg_ReconciliationtoNetIncomeLossAbstract_documentation_en-US" xlink:label="lab_kg_ReconciliationtoNetIncomeLossAbstract" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Reconciliation to Net Income (Loss) [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ReconciliationtoNetIncomeLossAbstract" xlink:href="kg-20250930.xsd#kg_ReconciliationtoNetIncomeLossAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_ReconciliationtoNetIncomeLossAbstract" xlink:to="lab_kg_ReconciliationtoNetIncomeLossAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_PeoActuallyPaidCompAmt_terseLabel_en-US" xlink:label="lab_ecd_PeoActuallyPaidCompAmt" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">PEO Actually Paid Compensation Amount</link:label>
    <link:label id="lab_ecd_PeoActuallyPaidCompAmt_label_en-US" xlink:label="lab_ecd_PeoActuallyPaidCompAmt" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">PEO Actually Paid Compensation Amount</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_PeoActuallyPaidCompAmt" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_PeoActuallyPaidCompAmt"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_PeoActuallyPaidCompAmt" xlink:to="lab_ecd_PeoActuallyPaidCompAmt" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_NumberOfNonIndependentDirectors_72d033ca-9201-4607-88cf-82dafe5d0fb3_terseLabel_en-US" xlink:label="lab_kg_NumberOfNonIndependentDirectors" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Number of non-independent director</link:label>
    <link:label id="lab_kg_NumberOfNonIndependentDirectors_label_en-US" xlink:label="lab_kg_NumberOfNonIndependentDirectors" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Number Of Non-Independent Directors</link:label>
    <link:label id="lab_kg_NumberOfNonIndependentDirectors_documentation_en-US" xlink:label="lab_kg_NumberOfNonIndependentDirectors" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Number Of Non-Independent Directors</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_NumberOfNonIndependentDirectors" xlink:href="kg-20250930.xsd#kg_NumberOfNonIndependentDirectors"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_NumberOfNonIndependentDirectors" xlink:to="lab_kg_NumberOfNonIndependentDirectors" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_AssetManagementAgreementMember_2e407c92-1842-4f7f-92b9-3073d2358ace_terseLabel_en-US" xlink:label="lab_kg_AssetManagementAgreementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Asset Management Agreement</link:label>
    <link:label id="lab_kg_AssetManagementAgreementMember_label_en-US" xlink:label="lab_kg_AssetManagementAgreementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Asset Management Agreement [Member]</link:label>
    <link:label id="lab_kg_AssetManagementAgreementMember_documentation_en-US" xlink:label="lab_kg_AssetManagementAgreementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Asset Management Agreement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_AssetManagementAgreementMember" xlink:href="kg-20250930.xsd#kg_AssetManagementAgreementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_AssetManagementAgreementMember" xlink:to="lab_kg_AssetManagementAgreementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RestrictedInvestmentNetOfLiabilities_6f50a711-fd41-4672-b80c-b05b31f40079_totalLabel_en-US" xlink:label="lab_kg_RestrictedInvestmentNetOfLiabilities" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Total restricted investments</link:label>
    <link:label id="lab_kg_RestrictedInvestmentNetOfLiabilities_label_en-US" xlink:label="lab_kg_RestrictedInvestmentNetOfLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Restricted Investment, Net of Liabilities</link:label>
    <link:label id="lab_kg_RestrictedInvestmentNetOfLiabilities_documentation_en-US" xlink:label="lab_kg_RestrictedInvestmentNetOfLiabilities" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Restricted Investment, Net of Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RestrictedInvestmentNetOfLiabilities" xlink:href="kg-20250930.xsd#kg_RestrictedInvestmentNetOfLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RestrictedInvestmentNetOfLiabilities" xlink:to="lab_kg_RestrictedInvestmentNetOfLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_FoundingShareholdersMember_2f24e7da-8343-428b-b1f4-04137ccbbf42_terseLabel_en-US" xlink:label="lab_kg_FoundingShareholdersMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Founding shareholders</link:label>
    <link:label id="lab_kg_FoundingShareholdersMember_label_en-US" xlink:label="lab_kg_FoundingShareholdersMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Founding Shareholders [Member]</link:label>
    <link:label id="lab_kg_FoundingShareholdersMember_documentation_en-US" xlink:label="lab_kg_FoundingShareholdersMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Founding Shareholders [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_FoundingShareholdersMember" xlink:href="kg-20250930.xsd#kg_FoundingShareholdersMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_FoundingShareholdersMember" xlink:to="lab_kg_FoundingShareholdersMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_RealEstateInvestmentsAtEquityMethodMember_3cecef7c-4ee5-41dc-8215-1a233be07323_terseLabel_en-US" xlink:label="lab_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Real estate investments at equity method</link:label>
    <link:label id="lab_kg_RealEstateInvestmentsAtEquityMethodMember_label_en-US" xlink:label="lab_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Real Estate Investments At Equity Method [Member]</link:label>
    <link:label id="lab_kg_RealEstateInvestmentsAtEquityMethodMember_documentation_en-US" xlink:label="lab_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Real Estate Investments At Equity Method</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:href="kg-20250930.xsd#kg_RealEstateInvestmentsAtEquityMethodMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:to="lab_kg_RealEstateInvestmentsAtEquityMethodMember" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_CashAndCashEquivalentsAndOtherMember_9355bf2e-63d8-4a00-a959-f09002aa4e9b_terseLabel_en-US" xlink:label="lab_kg_CashAndCashEquivalentsAndOtherMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash and cash equivalents</link:label>
    <link:label id="lab_kg_CashAndCashEquivalentsAndOtherMember_label_en-US" xlink:label="lab_kg_CashAndCashEquivalentsAndOtherMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash And Cash Equivalents And Other [Member]</link:label>
    <link:label id="lab_kg_CashAndCashEquivalentsAndOtherMember_documentation_en-US" xlink:label="lab_kg_CashAndCashEquivalentsAndOtherMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Cash And Cash Equivalents And Other [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_CashAndCashEquivalentsAndOtherMember" xlink:href="kg-20250930.xsd#kg_CashAndCashEquivalentsAndOtherMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_CashAndCashEquivalentsAndOtherMember" xlink:to="lab_kg_CashAndCashEquivalentsAndOtherMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock_97454f0c-d118-439c-8c33-efc60184bbb3_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reserve for Loss and Loss Adjustment Expenses</link:label>
    <link:label id="lab_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock_label_en-US" xlink:label="lab_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liability for Future Policy Benefits and Unpaid Claims Disclosure [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock" xlink:to="lab_us-gaap_LiabilityForFuturePolicyBenefitsAndUnpaidClaimsDisclosureTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss_dd35b637-6e5a-4109-bf48-d792126bf0bc_negatedLabel_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Fixed maturities, gross losses</link:label>
    <link:label id="lab_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss_label_en-US" xlink:label="lab_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Realized Loss</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss" xlink:to="lab_us-gaap_DebtSecuritiesAvailableForSaleRealizedLoss" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet_825fe4ad-3160-4fbc-bb14-3e01476fae1e_terseLabel_en-US" xlink:label="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net investment income and net realized and unrealized investment gains</link:label>
    <link:label id="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet_label_en-US" xlink:label="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment Income and Gain (Loss) on Investments, Net</link:label>
    <link:label id="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet_documentation_en-US" xlink:label="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Investment Income and Gain (Loss) on Investments, Net</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:href="kg-20250930.xsd#kg_InvestmentIncomeAndGainLossOnInvestmentsNet"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:to="lab_kg_InvestmentIncomeAndGainLossOnInvestmentsNet" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_AllIndividualsMember_terseLabel_en-US" xlink:label="lab_ecd_AllIndividualsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Individuals</link:label>
    <link:label id="lab_ecd_AllIndividualsMember_label_en-US" xlink:label="lab_ecd_AllIndividualsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Individuals [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllIndividualsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllIndividualsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllIndividualsMember" xlink:to="lab_ecd_AllIndividualsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_LongtermDebtTypeDomain_c20fd76d-5c63-4ac0-9b54-4586749e18d9_terseLabel_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Long-term Debt, Type [Domain]</link:label>
    <link:label id="lab_us-gaap_LongtermDebtTypeDomain_label_en-US" xlink:label="lab_us-gaap_LongtermDebtTypeDomain" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Long-Term Debt, Type [Domain]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LongtermDebtTypeDomain" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LongtermDebtTypeDomain"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LongtermDebtTypeDomain" xlink:to="lab_us-gaap_LongtermDebtTypeDomain" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueDisclosuresAbstract_596583b5-0075-4e2f-a954-33de5786db6e_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueDisclosuresAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair Value Disclosures [Abstract]</link:label>
    <link:label id="lab_us-gaap_FairValueDisclosuresAbstract_label_en-US" xlink:label="lab_us-gaap_FairValueDisclosuresAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value Disclosures [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueDisclosuresAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueDisclosuresAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_FairValueDisclosuresAbstract" xlink:to="lab_us-gaap_FairValueDisclosuresAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense_6bac78b2-656a-47f5-9362-bfeeb0e97bb0_negatedTerseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2009/role/negatedTerseLabel" xlink:type="resource" xml:lang="en-US">General and administrative expenses</link:label>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense_label_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, General and Administrative Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense" xlink:to="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationGeneralAndAdministrativeExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ForgoneRecoveryIndName_terseLabel_en-US" xlink:label="lab_ecd_ForgoneRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Name</link:label>
    <link:label id="lab_ecd_ForgoneRecoveryIndName_label_en-US" xlink:label="lab_ecd_ForgoneRecoveryIndName" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Forgone Recovery, Individual Name</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ForgoneRecoveryIndName" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ForgoneRecoveryIndName"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ForgoneRecoveryIndName" xlink:to="lab_ecd_ForgoneRecoveryIndName" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember_terseLabel_en-US" xlink:label="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Year-over-Year Change in Fair Value of Equity Awards Granted in Prior Years That are Outstanding and Unvested</link:label>
    <link:label id="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember_label_en-US" xlink:label="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Year-over-Year Change in Fair Value of Equity Awards Granted in Prior Years That are Outstanding and Unvested [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:to="lab_ecd_ChngInFrValOfOutsdngAndUnvstdEqtyAwrdsGrntdInPrrYrsMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations_848980c6-ee4c-4421-bcf3-d3868c79a7fa_terseLabel_en-US" xlink:label="lab_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net cash used in discontinued operations</link:label>
    <link:label id="lab_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations_label_en-US" xlink:label="lab_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Operating Activity, Discontinued Operation</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations" xlink:to="lab_us-gaap_CashProvidedByUsedInOperatingActivitiesDiscontinuedOperations" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EquityAwardsExpectedToVestOutstandingNumber_34bcc14f-7f5a-42ec-aaad-971130a0fd5a_terseLabel_en-US" xlink:label="lab_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares expected to be issued after vesting (in shares)</link:label>
    <link:label id="lab_kg_EquityAwardsExpectedToVestOutstandingNumber_label_en-US" xlink:label="lab_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Awards Expected to Vest, Outstanding, Number</link:label>
    <link:label id="lab_kg_EquityAwardsExpectedToVestOutstandingNumber_documentation_en-US" xlink:label="lab_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Equity Awards Expected to Vest, Outstanding, Number</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:href="kg-20250930.xsd#kg_EquityAwardsExpectedToVestOutstandingNumber"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:to="lab_kg_EquityAwardsExpectedToVestOutstandingNumber" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupClassificationAxis_fee9b5ab-6ea2-4e98-93af-9eea85cb7a18_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupClassificationAxis" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Disposal Group Classification [Axis]</link:label>
    <link:label id="lab_us-gaap_DisposalGroupClassificationAxis_label_en-US" xlink:label="lab_us-gaap_DisposalGroupClassificationAxis" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group Classification [Axis]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupClassificationAxis" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupClassificationAxis"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupClassificationAxis" xlink:to="lab_us-gaap_DisposalGroupClassificationAxis" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember_480c91b8-54a4-41c6-b9d7-3980e7d593fa_terseLabel_en-US" xlink:label="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">European Hospital Liability Quota Share</link:label>
    <link:label id="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember_label_en-US" xlink:label="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">European Hospital Liability Quota Share Agreement [Member]</link:label>
    <link:label id="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember_documentation_en-US" xlink:label="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">European Hospital Liability Quota Share Agreement [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:href="kg-20250930.xsd#kg_EuropeanHospitalLiabilityQuotaShareAgreementMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:to="lab_kg_EuropeanHospitalLiabilityQuotaShareAgreementMember" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract_a84515a8-d25c-4727-b3e0-5cc0b06577c1_terseLabel_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reconciliation of cash and restricted cash reported within Condensed Consolidated Balance Sheets:</link:label>
    <link:label id="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract_label_en-US" xlink:label="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash, Cash Equivalent, Restricted Cash, and Restricted Cash Equivalent, Continuing Operation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:to="lab_us-gaap_CashCashEquivalentsRestrictedCashAndRestrictedCashEquivalentsAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities_4beda440-198f-4c00-9296-e69f2167ff00_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Accrued expenses and other liabilities</link:label>
    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities_label_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Disposal Group, Including Discontinued Operation, Other Liabilities</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities" xlink:to="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationOtherLiabilities" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ErrCompAnalysisTextBlock_terseLabel_en-US" xlink:label="lab_ecd_ErrCompAnalysisTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Erroneous Compensation Analysis</link:label>
    <link:label id="lab_ecd_ErrCompAnalysisTextBlock_label_en-US" xlink:label="lab_ecd_ErrCompAnalysisTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Erroneous Compensation Analysis [Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ErrCompAnalysisTextBlock" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ErrCompAnalysisTextBlock"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ErrCompAnalysisTextBlock" xlink:to="lab_ecd_ErrCompAnalysisTextBlock" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_Rule10b51ArrTrmntdFlag_terseLabel_en-US" xlink:label="lab_ecd_Rule10b51ArrTrmntdFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Rule 10b5-1 Arrangement Terminated</link:label>
    <link:label id="lab_ecd_Rule10b51ArrTrmntdFlag_label_en-US" xlink:label="lab_ecd_Rule10b51ArrTrmntdFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Rule 10b5-1 Arrangement Terminated [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_Rule10b51ArrTrmntdFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_Rule10b51ArrTrmntdFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_Rule10b51ArrTrmntdFlag" xlink:to="lab_ecd_Rule10b51ArrTrmntdFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_EquityMethodInvestmentsInvesteeRevenue_076db8b7-a5c4-4072-8e4e-1f150de8c315_terseLabel_en-US" xlink:label="lab_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investee Revenue</link:label>
    <link:label id="lab_kg_EquityMethodInvestmentsInvesteeRevenue_label_en-US" xlink:label="lab_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Equity Method Investments Investee Revenue</link:label>
    <link:label id="lab_kg_EquityMethodInvestmentsInvesteeRevenue_documentation_en-US" xlink:label="lab_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Equity Method Investments Investee Revenue</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:href="kg-20250930.xsd#kg_EquityMethodInvestmentsInvesteeRevenue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:to="lab_kg_EquityMethodInvestmentsInvesteeRevenue" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_ErrCompRecoveryTable_terseLabel_en-US" xlink:label="lab_ecd_ErrCompRecoveryTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Erroneously Awarded Compensation Recovery</link:label>
    <link:label id="lab_ecd_ErrCompRecoveryTable_label_en-US" xlink:label="lab_ecd_ErrCompRecoveryTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Erroneously Awarded Compensation Recovery [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_ErrCompRecoveryTable" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_ErrCompRecoveryTable"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_ErrCompRecoveryTable" xlink:to="lab_ecd_ErrCompRecoveryTable" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_Security12bTitle_7fb1226d-66c3-4cfc-880d-3c8af1bb355c_terseLabel_en-US" xlink:label="lab_dei_Security12bTitle" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Title of 12(b) Security</link:label>
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    <link:label id="lab_us-gaap_SegmentReportingInformationLineItems_d5cbf0da-8eb7-4082-8d7a-58e573318338_terseLabel_en-US" xlink:label="lab_us-gaap_SegmentReportingInformationLineItems" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Segment Reporting Information [Line Items]</link:label>
    <link:label id="lab_us-gaap_SegmentReportingInformationLineItems_label_en-US" xlink:label="lab_us-gaap_SegmentReportingInformationLineItems" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Segment Reporting Information [Line Items]</link:label>
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    <link:label id="lab_kg_ReinsuranceAssetsAndLiabilitiesRetrocededPercentage_3638ced2-4abe-4157-866f-aac80b30c234_terseLabel_en-US" xlink:label="lab_kg_ReinsuranceAssetsAndLiabilitiesRetrocededPercentage" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Percentage of assets and liabilities associated with reinsurance business retroceded</link:label>
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    <link:label id="lab_kg_ReinsuranceAssetsAndLiabilitiesRetrocededPercentage_documentation_en-US" xlink:label="lab_kg_ReinsuranceAssetsAndLiabilitiesRetrocededPercentage" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Reinsurance, Assets And Liabilities, Retroceded, Percentage</link:label>
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    <link:label id="lab_us-gaap_LesseeOperatingLeaseTermOfContract_be06cdb3-9e14-41f2-8363-50b3fb3ba52e_terseLabel_en-US" xlink:label="lab_us-gaap_LesseeOperatingLeaseTermOfContract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Lessee, operating lease, term of contract</link:label>
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    <link:label id="lab_kg_ScheduleOfRestrictedCashAndInvestmentsTable_cb959d14-816a-4fe8-80f9-87b4a0ce0381_terseLabel_en-US" xlink:label="lab_kg_ScheduleOfRestrictedCashAndInvestmentsTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Restricted Cash and Investments [Table]</link:label>
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    <link:label id="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioCeiling_7708f4bc-04cc-49e0-b5f5-a84547f93f8e_terseLabel_en-US" xlink:label="lab_kg_RelatedPartyTransactionCommissionRateAdjustmentCriteriaLossRatioCeiling" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Commission rate, adjustment criteria, loss ratio ceiling</link:label>
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    <link:label id="lab_us-gaap_RetainedEarningsAccumulatedDeficit_c5b5bf58-0c7a-497e-bb12-0c85ad1d00a2_verboseLabel_en-US" xlink:label="lab_us-gaap_RetainedEarningsAccumulatedDeficit" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Retained Earnings (accumulated deficit)</link:label>
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    <link:label id="lab_kg_StandardPoorsBbbPlusBbbBbbMinusRatingMember_c3234918-cef8-42e8-86ff-c10e58f2045f_terseLabel_en-US" xlink:label="lab_kg_StandardPoorsBbbPlusBbbBbbMinusRatingMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">BBB+, BBB, BBB-</link:label>
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    <link:label id="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember_terseLabel_en-US" xlink:label="lab_ecd_YrEndFrValOfEqtyAwrdsGrntdInCvrdYrOutsdngAndUnvstdMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Year-end Fair Value of Equity Awards Granted in Covered Year that are Outstanding and Unvested</link:label>
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    <link:label id="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationForeignCurrencyTranslationGainsLosses_d887e1cb-4f20-45e0-b853-b0d2ad7e10bb_terseLabel_en-US" xlink:label="lab_us-gaap_DisposalGroupIncludingDiscontinuedOperationForeignCurrencyTranslationGainsLosses" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Foreign exchange and other losses</link:label>
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    <link:label id="lab_kg_EquityMethodInvestmentsRecordedWithEquityMethod_8b674dd6-6878-41c8-b783-eb409fd6cb83_terseLabel_en-US" xlink:label="lab_kg_EquityMethodInvestmentsRecordedWithEquityMethod" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Equity method investments</link:label>
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    <link:label id="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember_terseLabel_en-US" xlink:label="lab_ecd_EqtyAwrdsInSummryCompstnTblForAplblYrMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Aggregate Grant Date Fair Value of Equity Award Amounts Reported in Summary Compensation Table</link:label>
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    <link:label id="lab_us-gaap_StatementTable_02bcc3c8-987b-4bde-971e-88996e0d6216_terseLabel_en-US" xlink:label="lab_us-gaap_StatementTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Statement [Table]</link:label>
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    <link:label id="lab_us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock_09714111-622c-4ad7-94f5-e07cc4478ce7_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Purchase Price</link:label>
    <link:label id="lab_us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfBusinessAcquisitionsByAcquisitionTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Business Combination [Table Text Block]</link:label>
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    <link:label id="lab_kg_ProceedsFromMaturitiesPrepaymentsAndCallsOfDebtSecuritiesInvestingActivities_c93df809-8539-4942-9bb5-5363b1f72578_terseLabel_en-US" xlink:label="lab_kg_ProceedsFromMaturitiesPrepaymentsAndCallsOfDebtSecuritiesInvestingActivities" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Proceeds from maturities, paydowns and calls of fixed maturities</link:label>
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    <link:label id="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract_3529ccca-4ed7-4ecf-bc35-61be565fbd13_terseLabel_en-US" xlink:label="lab_us-gaap_AdjustmentsToReconcileNetIncomeLossToCashProvidedByUsedInOperatingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Adjustments to reconcile net income to net cash flows from operating activities:</link:label>
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    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_2c6188f2-678f-465f-bee4-1f2d19c69e2e_terseLabel_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Cash flows from financing activities:</link:label>
    <link:label id="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract_label_en-US" xlink:label="lab_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Cash Provided by (Used in) Financing Activity, Including Discontinued Operation [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_NetCashProvidedByUsedInFinancingActivitiesAbstract"/>
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    <link:label id="lab_kg_GainLossOnEquityMethodInvestments_ec4c0696-3a9f-470d-864a-e6a76ca1ea02_terseLabel_en-US" xlink:label="lab_kg_GainLossOnEquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Realized and unrealized gains (losses)</link:label>
    <link:label id="lab_kg_GainLossOnEquityMethodInvestments_label_en-US" xlink:label="lab_kg_GainLossOnEquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Gain (Loss) on Equity Method Investments</link:label>
    <link:label id="lab_kg_GainLossOnEquityMethodInvestments_documentation_en-US" xlink:label="lab_kg_GainLossOnEquityMethodInvestments" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Gain (Loss) on Equity Method Investments</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_GainLossOnEquityMethodInvestments" xlink:href="kg-20250930.xsd#kg_GainLossOnEquityMethodInvestments"/>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract_358a3271-a652-44f1-8664-844f8c728858_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amortized cost</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Amortized Cost, Fiscal Year Maturity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesAmortizedCostAbstract"/>
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    <link:label id="lab_us-gaap_InvestmentTypeExtensibleEnumeration_de0367bf-2333-4472-8c2b-748fb2dc1f12_terseLabel_en-US" xlink:label="lab_us-gaap_InvestmentTypeExtensibleEnumeration" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Investment, Type [Extensible Enumeration]</link:label>
    <link:label id="lab_us-gaap_InvestmentTypeExtensibleEnumeration_label_en-US" xlink:label="lab_us-gaap_InvestmentTypeExtensibleEnumeration" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Investment, Type [Extensible Enumeration]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_InvestmentTypeExtensibleEnumeration" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_InvestmentTypeExtensibleEnumeration"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_InvestmentTypeExtensibleEnumeration" xlink:to="lab_us-gaap_InvestmentTypeExtensibleEnumeration" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_d873d998-c10b-4f15-bb34-3d3514220bf2_terseLabel_en-US" xlink:label="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Reconciliation of Other Investments Measured at Fair value on a Recurring Basis Using Level 3 Inputs</link:label>
    <link:label id="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock_label_en-US" xlink:label="lab_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Fair Value, Assets Measured on Recurring Basis, Unobservable Input Reconciliation [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationTextBlock"/>
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    <link:label id="lab_ecd_AllAdjToCompMember_terseLabel_en-US" xlink:label="lab_ecd_AllAdjToCompMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">All Adjustments to Compensation</link:label>
    <link:label id="lab_ecd_AllAdjToCompMember_label_en-US" xlink:label="lab_ecd_AllAdjToCompMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">All Adjustments to Compensation [Member]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_AllAdjToCompMember" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_AllAdjToCompMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_AllAdjToCompMember" xlink:to="lab_ecd_AllAdjToCompMember" xlink:type="arc" order="1"/>
    <link:label id="lab_dei_AmendmentFlag_aaa06506-7b5b-45de-9777-b9588eb31e55_terseLabel_en-US" xlink:label="lab_dei_AmendmentFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Amendment Flag</link:label>
    <link:label id="lab_dei_AmendmentFlag_label_en-US" xlink:label="lab_dei_AmendmentFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Amendment Flag</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_dei_AmendmentFlag" xlink:href="https://xbrl.sec.gov/dei/2025/dei-2025.xsd#dei_AmendmentFlag"/>
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    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract_807795f4-0132-48a9-a712-9341a4e70409_terseLabel_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Fair value</link:label>
    <link:label id="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract_label_en-US" xlink:label="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Debt Securities, Available-for-Sale, Fair Value, Fiscal Year Maturity [Abstract]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract" xlink:to="lab_us-gaap_AvailableForSaleSecuritiesDebtMaturitiesFairValueAbstract" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized_ee2fec02-60d7-43ed-b8a7-e9788e7f6290_terseLabel_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Common shares issuable under incentive plans (in shares)</link:label>
    <link:label id="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized_label_en-US" xlink:label="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Share-Based Compensation Arrangement by Share-Based Payment Award, Number of Shares Authorized</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:to="lab_us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock_47832afe-313b-4164-8935-bd22eb30a1f9_terseLabel_en-US" xlink:label="lab_us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Intangible Assets Acquired</link:label>
    <link:label id="lab_us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock_label_en-US" xlink:label="lab_us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Acquired Finite-Lived Intangible Assets by Major Class [Table Text Block]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_ScheduleOfAcquiredFiniteLivedIntangibleAssetsByMajorClassTextBlock"/>
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    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_846f6bc3-8805-4cbb-80bc-8ab478ceee3b_verboseLabel_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Reserve for loss and loss adjustment expenses (includes $589,815 from related parties in 2025)</link:label>
    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_cd297ad7-41a0-49c2-ba51-d0b8421c5066_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Reserve for loss and loss adjustment expenses</link:label>
    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_81ade745-7968-4401-b8e3-a287c7374779_totalLabel_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/totalLabel" xlink:type="resource" xml:lang="en-US">Reserve for loss and LAE</link:label>
    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_0b36ab0a-4f65-42a4-b0bc-56e0be7de0d2_periodStartLabel_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/periodStartLabel" xlink:type="resource" xml:lang="en-US">Gross loss and LAE reserves, January 1</link:label>
    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_dca2fda5-ce6b-4bcb-b5f3-a064feb7f93b_periodEndLabel_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/periodEndLabel" xlink:type="resource" xml:lang="en-US">Gross loss and LAE reserves, September 30</link:label>
    <link:label id="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense_label_en-US" xlink:label="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Liability for Claims and Claims Adjustment Expense</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:to="lab_us-gaap_LiabilityForClaimsAndClaimsAdjustmentExpense" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_ScheduleofFixedMaturitiesTable_162da6a5-2bf6-417c-9173-9d7cc6a611a3_terseLabel_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTable" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Schedule of Fixed Maturities Table [Table]</link:label>
    <link:label id="lab_kg_ScheduleofFixedMaturitiesTable_label_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTable" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Schedule of Fixed Maturities [Table]</link:label>
    <link:label id="lab_kg_ScheduleofFixedMaturitiesTable_documentation_en-US" xlink:label="lab_kg_ScheduleofFixedMaturitiesTable" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Schedule of Fixed Maturities [Table]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_ScheduleofFixedMaturitiesTable" xlink:href="kg-20250930.xsd#kg_ScheduleofFixedMaturitiesTable"/>
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    <link:label id="lab_us-gaap_TreasuryStockCommonValue_5d719653-a7d8-45d8-b402-3aa4392c1be8_negatedLabel_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonValue" xlink:role="http://www.xbrl.org/2009/role/negatedLabel" xlink:type="resource" xml:lang="en-US">Treasury shares, at cost (2025: 2,237,534 shares)</link:label>
    <link:label id="lab_us-gaap_TreasuryStockCommonValue_label_en-US" xlink:label="lab_us-gaap_TreasuryStockCommonValue" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Treasury Stock, Common, Value</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_us-gaap_TreasuryStockCommonValue" xlink:href="https://xbrl.fasb.org/us-gaap/2025/elts/us-gaap-2025.xsd#us-gaap_TreasuryStockCommonValue"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_us-gaap_TreasuryStockCommonValue" xlink:to="lab_us-gaap_TreasuryStockCommonValue" xlink:type="arc" order="1"/>
    <link:label id="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember_bf2871e8-bd71-435b-944e-2805fe8a8f36_terseLabel_en-US" xlink:label="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Privately held preferred stocks</link:label>
    <link:label id="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember_4b784032-001a-43df-96d6-009f5ae278df_verboseLabel_en-US" xlink:label="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/verboseLabel" xlink:type="resource" xml:lang="en-US">Privately held equity investments - preferred shares</link:label>
    <link:label id="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember_label_en-US" xlink:label="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Privately Held Equity Investments in Preferred Stock [Member]</link:label>
    <link:label id="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember_documentation_en-US" xlink:label="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:role="http://www.xbrl.org/2003/role/documentation" xlink:type="resource" xml:lang="en-US">Privately Held Equity Investments in Preferred Stock</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:href="kg-20250930.xsd#kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:to="lab_kg_PrivatelyHeldEquityInvestmentsInPreferredStockMember" xlink:type="arc" order="1"/>
    <link:label id="lab_ecd_TrdArrTerminationDate_terseLabel_en-US" xlink:label="lab_ecd_TrdArrTerminationDate" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Termination Date</link:label>
    <link:label id="lab_ecd_TrdArrTerminationDate_label_en-US" xlink:label="lab_ecd_TrdArrTerminationDate" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Trading Arrangement Termination Date</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_TrdArrTerminationDate" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_TrdArrTerminationDate"/>
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    <link:label id="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag_terseLabel_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Adopted</link:label>
    <link:label id="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag_label_en-US" xlink:label="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:role="http://www.xbrl.org/2003/role/label" xlink:type="resource" xml:lang="en-US">Insider Trading Policies and Procedures Adopted [Flag]</link:label>
    <link:loc xlink:type="locator" xlink:label="loc_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:href="https://xbrl.sec.gov/ecd/2025/ecd-2025.xsd#ecd_InsiderTrdPoliciesProcAdoptedFlag"/>
    <link:labelArc xlink:arcrole="http://www.xbrl.org/2003/arcrole/concept-label" xlink:from="loc_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:to="lab_ecd_InsiderTrdPoliciesProcAdoptedFlag" xlink:type="arc" order="1"/>
    <link:label id="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent_d0aa8f2b-ce23-4699-bb76-ef6a40cf9fce_terseLabel_en-US" xlink:label="lab_us-gaap_OperatingLeaseWeightedAverageDiscountRatePercent" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Operating lease, weighted average discount rate</link:label>
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    <link:label id="lab_us-gaap_DeferredTaxAssetsOperatingLossCarryforwardsNotSubjectToExpiration_5ee5caf5-9e8a-4ea6-be0b-3f0bda40f3a2_terseLabel_en-US" xlink:label="lab_us-gaap_DeferredTaxAssetsOperatingLossCarryforwardsNotSubjectToExpiration" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Net operating loss carryforwards, with no expiry date, amount</link:label>
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    <link:label id="lab_kg_PrivatelyHeldEquitySecuritiesInCommonStockMember_29e953ee-ae11-459a-ab6e-6c84908a98ee_terseLabel_en-US" xlink:label="lab_kg_PrivatelyHeldEquitySecuritiesInCommonStockMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Privately held common stocks</link:label>
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    <link:label id="lab_kg_A2019OmnibusIncentivePlanMember_6a8a36d7-254b-431d-b181-a64654b0d58a_terseLabel_en-US" xlink:label="lab_kg_A2019OmnibusIncentivePlanMember" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">2019 Omnibus Incentive Plan</link:label>
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    <link:label id="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseNetAbstract_5218c7be-988c-4463-b301-90ee3101df77_terseLabel_en-US" xlink:label="lab_us-gaap_LiabilityForUnpaidClaimsAndClaimsAdjustmentExpenseNetAbstract" xlink:role="http://www.xbrl.org/2003/role/terseLabel" xlink:type="resource" xml:lang="en-US">Liability for Unpaid Claims and Claims Adjustment Expense [Roll Forward]</link:label>
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