Honestly, get outta London. There's some cool things to do there but mostly it's an insanely overpriced tourist trap. If you want to go there and spend shittons of money on haute couture, that's cool. I'd suggest heading up to Scotland or to Ireland if you want to soak up some real culture.
Bring comfy shoes, you'll be walking a lot.
Have a dummy wallet with some BS membership/expired cards to fool pickpockets. Carry only the essentials in your real wallet; have a phone card with some money on it, a CC (Capital One has no international transaction fees IIRC), ID, and a little cash. Always have a little cash just in case.
Don't wear shorts unless you want to stick out like a sore thumb.
Know that typical English food = curry.
Do at least one Michelin rated restaurant experience. My sis said Restaurant Gordon Ramsay was amazing but she was working for him at the time so she's biased, and Alain Ducasse's namesake restaurant is also a 3-star experience and he's one of the best chefs in the world.
Of all the touristy things you might do, definitely go to The British Museum. Lot of human history and culturally relevant things there, I thought it was actually really cool.
Travel is best accomplished by train, it's quite comfortable.
I'll update as I think of things.
OH if you're not going to be traveling and would rather just stay in London, look into AirBnb or some other service to rent an apartment for a week. Way better way to immerse yourself in the city. That way if there's a day or two you'd like to skip tourist things and just get some fresh bread/cheese/meats and stay in, it's in a nicer environment than a hotel room.
Source: my own travels, my sister lived there for a year, best friend's gf just returned from a 2-year work assignment.