Fly play from Stansted to Baltimore is under 1000 pounds, Boston is 1600 and NY for under 1000 sterling, but I would probably avoid the NY route as it's way out of the city at a small airport called Stewart. Baltimore airport is actually not too far from Washington and you can get the train if you don't want to drive. Weather in September will have lost its intense heat so it's a great time to visit.
I would HIGHLY recommend Washington for that age, it's fantastic, easy to get around, the Smithsonian Museums are free, the food is great and watching Secret Service snipers on the roof of the White House at the W hotel is great. Arlington Cemetery with the Kennedy grave is great and honestly it's like movie. You can get public transport out to Mount Vernon, Washington's place too.
We've stayed in Rosslyn as it's right on the underground station, at the Holiday Inn, they do a free (at least they used to ) breakfast on a roof top restaurant overlooking Reagan Airport. Homewood Suites are good as they tend to give you a kitchenette as well as being a bit more spacious with teenagers. Sonder will be good as well, a bit more space for a week but bear in mind they are functional, clean and not terribly fancy. Other areas to consider is Georgetown (where the diplomats tend to live).
Check out the Founding Farmer's restaurant and make sure you book White House, Congress Library before you come. Haven't been to Boston but my husband works there sometimes and loves it. New York would be third on my list of those three places, I'm really not a fan, too unfriendly, expensive and dirty these days.