Any particular things you're interested in generally? Asking "What should I do in London?" is kind of like asking "What should I do in NYC?" There are a zillion things to do, so it all just depends on what you're into. A few ideas:
- If you just want to see the sights and sounds, I actually think hop on hop off bus tours are a good way to do it quickly. Yeah, they're touristy, but you're a tourist.
- Kind of in the same vein, the London Eye can be a good way of just being able to look over the whole city. Much shorter experience, though.
- If you do it on your own, the main sights that come to mind are Buckingham Palace, the Tower Bridge, the Tower of London, Big Ben, and Westminster Abbey. Depending on the lines, the Abbey and Tower of London are worth going through in my opinion.
- In terms of museums, the British Museum is free and worth the trip if you're into all the ancient stuff the Brits stole from the places they conquered. The Natural History Museum is good and free, as well if you're into dead animals.
- Like NYC, theatre is a big deal if that's your thing. Tons of shows to see every night. Ask if you want recommendations.
If you're trying to stay affordably, the biggest thing to me is getting out of the central city a bit, but stay on a tube (subway) line. Once you're in the central city, getting from place to place isn't bad if you can walk a bit, and you can always do taxis if you need to (though the costs add up after a while).
Again, it all really just depends on what you like to do, so ask if you have specific questions.