Yay! We sort of did it, without losing too many resources. Now to get the snoozing guy... somehow... perhaps non-fatally?
@WH For London, the centre is pretty tightly packed with all the cool land marks and so forth. Added bonus there will be Shaun the Sheeps around to go look at!
http://shauninthecity.org.uk/
Anyway, the best bet if you have the money is to get one of the tour buses like the Original Tours. They basically give you a trip around the cioty for 2 - 3 hours to get a feel for everything around, then you hop on hop off to actually go to them. You can get a two day ticket so that you have a day and a half to explore what they point out.
The south bank of the Thames tends to be good to walk along just for the variety of shops and sights of London. If you go West along there you can actually hit Battersey park and see the old station.
Otherwise, stick to the centre and see the Tower of London, Tower Bridge, Westminister, all that fun stuff :)
Actually, all the thames shuttles between the two piers tend to do voluntary commentaries so you can learn some history from them too and also see the city from the river.
Also Trafalgar Square and the gallery leads to Leicester Square and all the food places too :)
If you more a museum buff then the natural history and british museum are cool, if you like art both the national gallery and also on the south bank the tate modern is cool (that is on the opposite side of the river to St Paul's, and you can cross the millenium bridge to them)
You might be able to spot my mum has to go through London when she comes down, so we end up spending some time in London :)