It depends.  There are some states (NJ being one) that does require you to file both a part-year and a nonresident return for the same tax year.  Ohio does not do this, however, so all of your income you earned working in Ohio is taxable in Ohio.  (Normally KY and OH are reciprocal, but reciprocity is cancelled when it is a part-year return).  Because of tax rules and how the programming works, I would file this way:

1. KY nonresident return.  It is true that technically you file a part-year return to KY, but this will be accurate for a couple of reasons.  In the first place, KY calculates your tax the same for a Nonresident or a Part-Year Resident.  Second, since you were likely paying (or should have been paying) KY tax on this income, this will represent your situation tax-wise accurately.  Although the reciprocal rules are not in effect, Ohio gives you a credit for the tax you pay to KY for the income earned in the "reciprocal" situation.  Third, it is very difficult to get TurboTax to do three part-year returns.  The easiest way to get this to work is to claim one state nonresident if you can justify it, and in this case, you can.
2. Ohio part-year return.  Use Jan. 1 as your residency start date and your move to Illinois as your move-out date.  Make sure that you are credited for taxes you paid to KY on the income "earned" there.
   
3. Illinois part-year return.  This you would have done for Illinois regardless.
   

Here is an FAQ to assist you with the preparation of the nonresident return:  https://ttlc.intuit.com/replies/3302052