In all honesty the best flight route between Washington DC and London is on Air Force 1, but considering that idealo has not quite managed to secure a deal with that airline, there are a few very good plan B options. British Airways, United and Virgin Atlantic will take you the 3 and half thousand miles across the pond in just over 7 hours. And with between 7 and 8 direct flights a day to choose from, you might just find yourself feeling somewhat presidential as you swoop over Buckingham Palace.
The best way to get a low cost air ticket would be to consider having a few stops on your trip. Airlines that offer good deals with one stop are American and Air France. But if you plan well in advance there often is little price difference between direct flights and ones with stop overs, so booking early remains your best way to save money.
Regardless of which airport you choose to land in, although the most likely would probably be Heathrow Internationl Airport if you are on a direct flight, getting around the city will not be a problem. It just so happens that London has the oldest subway network in the world, so they Brits are practiced in getting you were you need to be fast and in old world style.