The fair place to meet is Oklahoma City, OK — the city closest to the midpoint of Las Vegas and Louisville. From the farther side that’s about 20 hr 42 min of driving.
Recommended midpoint
Oklahoma City, OK
From Las Vegas
20 hr 42 min
987 mi to Oklahoma City
From Louisville
14 hr 12 min
677 mi to Oklahoma City
Louisville has the shorter trip; the split is off by about 6 hr 30 min. The alternatives below can even it out. Las Vegas and Louisville are about 1,616 miles apart.
Las Vegas and Louisville are about 1,616 miles apart by road. Split the difference and you arrive near Oklahoma City, the city closest to the halfway point between them. That puts roughly 20 hr 42 min of driving on the Las Vegas side and 14 hr 12 min on the Louisville side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Oklahoma City still makes a fair, central place for Las Vegas and Louisville to converge, splitting the travel instead of asking one side to cross the country.
If Oklahoma City doesn't have what you're after, Denver and Kansas City are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Las Vegas and Louisville.
Oklahoma City, often shortened to OKC, is the capital and most populous city in the U.S. state of Oklahoma. It is the 21st-most populous U.S. city and 8th largest in the Southern United States, with a population of 681,054 at the 2020 census. Read more →
Las Vegas, colloquially shortened to Vegas, is the most populous city in the U.S. state of Nevada and the county seat of Clark County. It is the 24th-most populous city in the United States, with 641,903 residents at the 2020 census. 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.