Q. Can she claim she was a Kentucky full time resident and file only KY tax return?
A. No. She must file an IL resident return.
Q. If not, what should be done to transfer the already payed state income taxes from KY to IL?
A. That can't be done. She files for a refund from KY, Then pays tax to IL. She may even owe a late payment penalty.
KY has a specific form (740 NP-R ) for a reciprocal state refund. Rather than paying TurboTax for KY software, you should try filing that form by hand. It looks pretty simple.
https://revenue.ky.gov/Forms/740-NP-R.020521.pdf
In the personal info section, when it asks if you made money in another state, answer NO, per the reciprocal state instructions, on that page. This will prevent TurboTax from generating the KY program.
The reciprocity agreement does not apply to local city or county income taxes. You are not required to file a local KY return, but you are also not entitled to a refund of the city/county withholding (box19 on your W-2), unless there was an error in the amount.
Her employer may be unwilling to withhold IL tax, but she should ask them to stop withholding KY state tax. Give them KY form Form 42A809
