The fair place to meet is Indianapolis, IN — the city closest to the midpoint of Kansas City and Louisville. From the farther side that’s about 9 hr 28 min of driving.
Recommended midpoint
Indianapolis, IN
From Kansas City
9 hr 28 min
452 mi to Indianapolis
From Louisville
2 hr 40 min
107 mi to Indianapolis
Louisville has the shorter trip; the split is off by about 6 hr 48 min. The alternatives below can even it out. Kansas City and Louisville are about 479 miles apart.
Kansas City and Louisville are about 479 miles apart by road. Split the difference and you arrive near Indianapolis, the city closest to the halfway point between them. That puts roughly 9 hr 28 min of driving on the Kansas City side and 2 hr 40 min on the Louisville side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Indianapolis suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Indianapolis doesn't have what you're after, Memphis and Nashville are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Kansas City and Louisville.
Indianapolis, colloquially known as Indy, is the capital and most populous city of the U.S. state of Indiana and the county seat of Marion County. Indianapolis is situated in the state's central till plain region along the west fork of the White River. Read more →
Kansas City, abbreviated KC or KCMO, is the largest city in the U.S. state of Missouri by both population and area. It is located on the Missouri River at its confluence with the Kansas River, within Jackson, Clay, Platte and Cass counties. Read more →
Louisville is the most populous city in the U.S. state of Kentucky, sixth-most populous city in the Southeast, and the 27th-most-populous city in the United States. Read more →
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.
Estimates use straight-line distance and typical road speeds; real drive times vary with route and traffic.