Txgunrnnr said:
--Windsor Castle
--Tower of London & Tower Bridge
--The Shard
--Wimbledon
--Changing of the Guards
--Westminster Abbey
--Churchill War Rooms
--Greenwich ( Cutty Sark )
These are all great - one bit of advice. The best place to see the changing of the guard is actually at Windsor Castle as compared to Buckingham Palace (I think its Tue/Thur/Sat, but verify). You can get so much closer and see everything.
We watched the change then toured the Castle. Came out and they were changing the individual guard in one of the guard spots. Watched that and could have easily walked with the detachment as they walked to the next guard post.
Other spots:
Borough Market
National Gallery
British Museum
Imperial War Museum
Victoria & Albert Museum
National Portrait Gallery
Wallace Collection
John Soames House
National History museum
There is just so much to do in London that you could stay there for a week and not do it all.