Just returned from a week in London. What a world-class city!
We stayed in an aparthotel in the Whitechapel area: The Gate Hotel.
It's a great location for catching the subway (Tube) and within walking distance of Brick Lane (London's curry district) and Tower Bridge if you're so inclined. Get a little Tube map and hit the Aldgate East station and the city is basically your oyster. And speaking of oyster, you don't need to get an Oyster card. Just use your credit/debit card and tap in and tap out. As easy as falling off a log.
Some people are the museum types, but I don't know how anyone can go to London and not see the Rosetta Stone and the Elgin Marbles (Parthenon statues now) in the British Museum. Free entry, spend an hour or so and Bob's your uncle. The National Portrait Gallery and National Gallery are also pretty good. We had the great fortune of having a friend who's a local and he helped guide us around.
Here's my take on some of the places, of course, ymmv:
British Museum: worth it.
Notting Hill: NOT worth it.
Trafalgar Square and environs (around the Embankment Tube station): totally worth it.
Brick Lane (street near our hotel) and the weekend market (mostly food stalls): worth it at least once.
Borough Market: mind-blowing but you can't eat at every food stall and you probably don't need to do your weekly grocery shopping there, but worth it.
Jack the Ripper tour, Westminster tour (both with guides) TOTALLY worth it.
Bus tour to Windsor Castle, Stonehenge, and Bath: eh, it's ok, but the downside is you get to Bath and have an 75 minutes to see things. It's a trade-off imo, quantity over quality. I liked Windsor Castle, Stonehenge was meh, and Bath was a notch or two above meh.
Westminster Abbey: good, but not great imo. It was kinda dark that day and there's room after room (or alcove after alcove) of what I guess were famous people I've never heard of with a bunch of stuff written in Latin about them. That stuff gets old fast, imo. Also, the audio guide was not good. But I did like the Scientists "corner" (Isaac Newton, Stephen Hawking, Paul Dirac, dudes like that, and the Poets' Corner. But even so, it's essentially a bunch of names on the floor. Worth seeing but don't put it on the top of your list, imo.
Hampton Court Palace: Henry VIII's place. We had a tour guide and it was fantastic. TOTALLY worth it. Better than Windsor Castle and Westminster Abbey. Plus, there's a nice little pub nearby, The Mute Swan, that has good beer and good food. Speaking of pubs, we went to the Blind Beggar (favorite haunt of the Kray brothers - look them up if you haven't heard of them, catch a movie about them, NOT the Tom Hardy one but the 1990 movie "The Krays") on Friday night and it was basically a so-so college bar with a bunch of drunk 20-year olds. NOT how an American would envision an English pub. Same goes for the White Hart on Whitechapel High Street. Other than a sign that reads 1729 (iirc) it was just another hangout for slightly older drunks. However, the Churchill Arms in Notting Hill was good.
Now, don't laugh too much, but we caught the ABBA Voyage show in the ABBA arena (specially built for the show) on Pudding Mill Lane, and it was incredible. Just incredible. Basically holograms of ABBA (but with a live band). TOTALLY worth it even if you're not a big ABBA fan. Depending on your views of hallucinogens you might want to imbibe a bit before the show. But seriously, I would consider the ABBA show a must see.
We walked about 8 miles a day and took the tube basically everywhere.
Go, spend a week in a central location, and enjoy yourself. Cheerio mate!