London eh? hmmm well I've spent most of my spare time on this planet in or around London. However as a local I don't do any touristy things.
There are plenty of museums in Kensington also Royal Albert Hall may have something on.
Harrods in Knightsbridge if you're feeling rich
Orcs Nest on Charring Cross Road (get out at Tottenham Ct Road tube and walk down)
Leisure Games in Finchley Central (Northern Line Tube)
Shopping - Oxford St, Regents Street, Bond St, Tottenham Court Road, etc... But its gonna be cheaper in the US mate.
Camden is a cool place but not for everyone.
Depending on what you are looking for London probably has it! Watch your belongings and pass your blend in check please!
Take morrolan's advice and get out of the city too. If you have time Stonehenge is about a 2 hour drive from central London
Somerset/Devon is about 3-4 hours drive from central London - Best coast in my opinion is Wollacombe in NW Devon. Best scenery is found all over Somerset - truely breathtaking along the A39 Coastal Road.
Roman Historical town of St. Albans down the road from me (and a bloody good night out) about 30 mins away on the Thameslink from Kings X.
Errr loads more but as I say I don't do it

A good day in London for me is:
Visit large Virgin Megastore, HMV, GW in the Plaza. Into Borders on Oxford St. for a caramel frappachino and a chat/read (its cool grab a book and sit down with it for a bit before buying) nip down to Orcs Nest for the gaming stuff then onto China Town (off Leicester Square) for an all you can eat buffet lunch. Have a wander around Leicester Sqare to see if any films are on and a sit down on the green (in the summer) then into Lillywhites Sporting goods (hella busy!). Onto Covent Garden for a look to see if anything decent is going on (usually not) maybe have a pint in the Punch and Judy there then back home or to Leicester Sqaure for a film. In the evening there are loads of gigs on - Marquee, Astoria, Mean Fiddler, Dublin Castle, Worlds End, etc...
Have a great time!