The fair place to meet is Kansas City, MO — the city closest to the midpoint of Denver and Washington. From the farther side that’s about 19 hr 43 min of driving.
Recommended midpoint
Kansas City, MO
From Denver
11 hr 41 min
557 mi to Kansas City
From Washington
19 hr 43 min
941 mi to Kansas City
Denver has the shorter trip; the split is off by about 8 hr 2 min. The alternatives below can even it out. Denver and Washington are about 1,489 miles apart.
Denver and Washington are about 1,489 miles apart by road. Split the difference and you arrive near Kansas City, the city closest to the halfway point between them. That puts roughly 11 hr 41 min of driving on the Denver side and 19 hr 43 min on the Washington 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. Kansas City still makes a fair, central place for Denver and Washington to converge, splitting the travel instead of asking one side to cross the country.
If Kansas City doesn't have what you're after, Chicago and Indianapolis are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Denver and Washington.
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 →
Denver is the capital and most populous city of the U.S. state of Colorado. Officially a consolidated city and county, it is located in the South Platte River valley on the western edge of the High Plains, and is just east of the Front Range of the Rocky… Read more →
Washington, D.C., officially the District of Columbia and commonly known as simply Washington or D.C., is the capital city and federal district of 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.