+1 for Edinburgh. I went there on a day-trip but immediately wished it had been a week trip.
Of course I recommend the Natural History Museum where I was working for the last 4 months! It and the science museum directly next door are excellent.
I got a real kick out of the Cabinet War Rooms / Churchill bunker tour. It was a little pricier than a lot of the other museums (which are frequently free), but it's in remarkably original condition and was fascinating to see.
On the subject of WWII, I found it extremely fascinating to see bomb damage to the NHM and V&A (Victoria and Albert museum) that they left as a reminder. Just around to the Exhibition Road side of the NHM near the entrance on that side and you'll see huge chunks of the building stone missing from a bomb that landed right in the middle of Exhibition road there.
Camden Market was a madhouse, but interesting just to see. If you get back into the old horse stables area I think it's especially cool.
Kew Gardens is not very far out of London and was really amazing. I was dragged to a lot of gardens by my parents when I was young and hated it, but I actually thought Kew was cool.
Windsor Castle tour gets a "OK" rating. If you're into weapons then it'd be great because they have so many old ones lying around they started making artistic patterns with them on the walls.
I strongly recommend walking along the Thames all the way from Westminster Bridge along the south bank to at least Millennium Bridge / The Tate Modern, or as far as Tower Bridge if you like walking.
I also recommend walking from Covent Garden down through Leicester Square and on to Piccadilly Circus. It's tourist city but for a reason.
If the weather's nice, I went on one of these famous walks and enjoyed it:
http://www.walks.com/