What amount do I put in for Part-Year and Nonresident Income Allocation, Kansas amount?

Yes.  KS does not have a tax reciprocity agreement with any states therefore any income you earned in KS is taxable by KS.  

You didn't mention your resident state, but if that state has an income tax, meaning you are filing a resident tax return, you would claim a credit for taxes paid to another state on that resident return.  

Your resident state taxes all income from all sources so the KS earned income is double taxed.  To mitigate that double taxation your resident state offers the credit for taxes paid to another state.

In TurboTax if you work your non-resident state interview first, information used to claim the credit for other state taxes automatically flows to your resident interview.