Even better idea: Dont go to london at all!
Now that's one of your worst ideas, I think.
You know, I've lived near to London for much of my life. I've walked around it thousands of times, I learnt to drive there. I used to cycle and motor-cycle around it for years.
But I've only seen about 1% of it, I think.
Some other things:
Take one of the evening "ghost tours". If you can get one that goes around Whitechapel (hunting ground of Jack the Ripper), they can be quite scary. But there are many others. I've been on a few - great fun and a pleasant evening walk.
If you go to London and arrive at Victoria Station, it's a short walk to Buckingham Palace (where Mrs King lives). They allow visitors into the Palace these days, so you could go around there. After that, a short walk up the Mall, past the Serpentine to Trafalgar Square, where you can see the lions and Nelson's Column and chase the pigeons (everyone does).
From there, you can catch one one of the "round London" tour buses. They go around the Capital, stopping at lots of different places. You can get on and off at lots of different points, travelling around London City in a circle.
Get off at London Dungeons and see the medieval torture equipment (bring some back for me to use on some people on the forum)
Get off at London Eye and see the whole of London as you rise into the air, above the Houses of Parliament and the Thames.
The River buses are on the other side of the bridge.
Remember that London is only a 1 mile square around Moorgate area. What most people call "London" is actually a different city, namely "Westminster". Then, the county of London is immense. It's about 30-40 miles across.