You do not owe DC tax for unemployment received after you moved to KS.
You do not owe KS tax for unemployment received while you lived in DC.
If you were still a resident of DC while receiving unemployment based on your DC job, then you would file a part-year resident return for DC. On that return, report any income you earned while in DC, including your unemployment. If DC tax was withheld (wages or unemployment) for that period, you may qualify for a refund (or, depending on your total income, may owe additional DC tax).
Any income you received as a resident of KS (unemployment from your DC job or KS employment) is subject to KS tax. File a part-year resident return for KS to report that income.
