Elkhorn Creek is the closest thing to a real whitewater run near Louisville, it is near Frankfort, KY about a one hour drive away. It is mostly Class I/II, with maybe a Class III on the upper section at higher water levels. Before planning a trip you will want to check the USGS Gauge for the Elkhorn.
(Often referred to as the "whitewater section"). The USGS gauge needs to be at least 300cfs for the upper section unless you don't mind dragging. Optimal levels for whitewater are around 2500-3000cfs, above 3000cfs the portage at the dam becomes more difficult. American Whitewater has a good writeup of the upper section.
There are now several access points to the upper section of the Elkhorn. The usual one is at the Forks of the Elkhorn Campground, and costs $2 per boat to launch. There is another (new) put-in just East of the campground which is free. Another access point, previously referred to as the "illegal takeout" is on Peaks Mill Rd about half way to Elkhorn Acres, you will see a brown sign on the side of the road marking it, but this spot has very limited parking. The last one is Elkhorn Acres, and is owned by American Whitewater which is free to use.
The lower section of the Elkhorn starts at the takeout for the upper section and runs to the 127 bridge. This is mostly class I with a couple of class II rapids at good levels. There are no generally accepted cut-off levels, but if the USGS gauge is below 200cfs, plan on some dragging.
It is generally accepted that it takes about 24 hours for rain to propagate down to the "upper section" of the Elkhorn. The University of Kentucky has maps of the watersheds of the two forks which come together to form the upper section.
Canoe Kentucky Rents kayaks and canoes on the lower section, and sometimes offers shuttle service for the upper section. They also often offer trips on the Kentucky River during periods of low water on the Elkhorn.
There are a few places to camp on the banks of the upper section, but they aren't often used due to the fact that it's a short run. Possibly the most interesting one, for those interested in surfing, would be the island in between the Surf City and Angeoplasty rapids.
Shuttle to put-in at forks of the Elkhorn
Shuttle from Elkhorn Acres to 127 bridge:
Alternate takeout on Peaks Mill Rd