You’re coming to London for a reason. You must at some time have said “I really want to see/do XXX”.
When you arrive in the UK, at Heathrow say, it’s not impossible some immigration person will ask you what you are coming for, what are you going to do while you are here. They always ask us in the USA. You don’t have to do what you say, no-one will check on you. They just want to know you have sensible plans.
If you say “go up the Eiffel Tower”, you have problems. Wanting to see Buckingham Palace is just fine (just know it is massively disappointing and you won’t see the King).
It would be a shame to come here with a list of things you wanted to do, and not do most of them, so get yourself organised and make a simple itinerary. And yes, London is great for just ambling around just seeing what’s what, and you can do a bit of that too, but make a list.
We can’t do it for you. We don’t know what you are interested in. Later, we do have suggestions of things to do, but they may not appeal.
Also, if you are coming to eg see a play, go to a museum, get tickets in advance if you can. Using foreign credit cards on-line may be a pain.
Now, some museums are free and you can just walk in, but some are not and special exhibitions are often charging and will be busy.
Plus, buying tickets in advance can give your holiday some structure.