I've been kayaking around the New Albany/Jeffersonville/Charlestown waterways like the Ohio River, Silver Creek and 14 Mile Creek, but wanted to tap the INGO knowledge base for any suggestions for a good paddle. Travel is okay.
Big South Fork of the Cumberland River from Leatherwood Ford in Tennessee back to Lookout in Kentucky. It's a two day trip (+/- 30 miles), some whitewater, awesome scenery, great fishing, almost zero people. There is virtually no river access from the start to finish. It is easily do-able in a canoe. Did it with my wife many years ago. Camp on the river wherever you want. There is an outfitter in Tennessee who will ferry you back and forth for a reasonable fee.
You should check out IYAK - Indiana Kayakers (Indianapolis, IN) - Meetup if you like to kayak with others. I've been on a few different trips with them now & had a good time every time. There may be more groups closer to you as well.