Umm...
I've actually forgotten what I did in London when I went there a couple of years back.
There was sitting outside Buckingham Palace because my family wanted to see the changing of the Guard.
Umm...
There was a tour of the Tower of London, which was kinda interesting if you're into history, otherwise its largely meh beyond a couple of suits of armour.
There was some London Bridge thing, and some other similar thing somewhere else, where it tried to be one of those spooky walk through exhibits on the history of London with people following you, or jumping up behind you, and things looking like they were going to fall from the ceiling and such. In one of them there was a funny judge who pronounced everyone guilty for a variety of amusing reasons, but that might have been at Warwick Castle, I'm not sure.
There's the Globe Theatre [Or reconstruction thereof] if you're into Shakespeare. I think we saw McBeth there, and it wasn't too expensive [Though I could be wrong].
Eh, most of what my family did in England was all over the place - Dover, London, wherever Stonehenge is, Cambridge, Stratford Upon Avon, Warwick Castle, some other places I've forgotten the name of - nothing too far from London, but still occasionally a couple of hours drive. If you get bored whilst in London, and Daystar is too busy for a duel, I'd take a drive out to see some of the other places in England too.
And if you're in the Northern Hemisphere whereby your Summer is Australia's Winter, dependent on when you go next year [As it would be when Summer comes around in the Northern Hemisphere], I might be able to run into you, as I'll be touring with my choir over there for some music Festival in Wales before heading down to London, then across to Paris.