IL has a reciprocal agreement, on wages, with KY. No KY state taxes are withheld or due and you do not need to file a KY return.
However the reciprocity agreement does not apply to local city or county income taxes. You are not required to file a local return, but you are also not entitled to a refund of the city or county withholding (box19 on your W-2), unless there was an error in the amount. IL will give you a credit for any local KY tax paid/withheld.
