I see JustaShooter has more or less answered your question. The bad news is that it'll cost you about $150. The good news is that you can handle that with one day of classroom + range time, and it'll take about a month at worst. Meantime, if your KY license is still valid, it's good here for a while.
(I haven't looked up the costs in a while, so don't hold me to that figure, but it's pretty close.)
I think you'll still need a drugstore passport photo with your application (Walgreens did a fine job on my last renewal), but that's about the worst of the extra stuff. (Lawrence County PA, where I renewed my PA license in May, requires two LOCAL references. Not a problem for me, but not everybody lives so close to PA that an errant shot from my club's outdoor range can land in PA.
Each Sheriff does things a little differently, but I used Columbiana County, and their procedure is:
1. Visit the "Downtown" Sheriff's Office to drop off your application and give 'em your money.
2. Wait about a week and go to the Sheriff's Station at the Jail (way outside of town) to start the paperwork.
3. Wait another week and go to the Jail again to finish the process and walk out with your new license.
For an in-state renewal, steps 2 and 3 could be handled together in about an hour (of mostly waiting), but they severely restrict the availability of the Deputies who handle this. Some counties don't.... A new application from somebody who's just moved into OH takes a little longer.
There's a sticky at http://forums.buckeyefirearms.org/viewtopic.php?f=6&t=10097
that you may want to have a look at, but it's not mandatory and may be hazardous to your health. Just a bit about how this place works, and who's here. You don't need to go past the first couple pages.