Q. Can I get the KY local taxes back?
A. No. The reciprocal agreement between IN & KY on state taxes does not apply to local city/county taxes. You are not required to file a KY local city return, but you are also not entitled to a refund of the city/county withholding (box19 on your W-2),
Most IN counties (but not the state) allow a credit, or partial credit for tax paid to an KY city. That will come up in the IN state interview. (For details, see:
You do not get the tax back but non-Indiana local tax may qualify for a Credit for Local Taxes Paid Outside of Indiana. See page 36 of the Indiana 2023 IT-40 Instruction Booklet. See here.
Credit for Local Taxes Paid Outside of Indiana
If you figured county tax on Form IT-40, line 9, and had to pay a local income tax outside Indiana, you may be able to take a credit. This credit applies only if the tax you paid outside Indiana was to another city, county, town, or other local governmental entity; and they did not refund the tax, or give you a credit for Indiana county tax.
In TurboTax Online, the credit is computed at the screen Take a look at Indiana credits and taxes. Click Start / Revisit to the right of Credit for Non-Indiana Local Taxes.