Miami and Washington are about 927 miles apart by road. Split the difference and you arrive near Charlotte, the city closest to the halfway point between them. That puts roughly 13 hr 44 min of driving on the Miami side and 6 hr 55 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. Charlotte still makes a fair, central place for Miami and Washington to converge, splitting the travel instead of asking one side to cross the country.
If Charlotte doesn't have what you're after, Raleigh and Orlando are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Miami and Washington.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.