Museums - take your pick, Most of them probably have admission charges, but they'll be worth it.
British Museum (basically stuff we pillaged in our Empire days)
Natural History Museum
Science Museum
Victoria and Albert Museum. (art)
Maritime Museum (Greenwich)

Places to see:
Buckingham Palace
Downing Street
Westminster
etc, etc, etc, etc...

Best advice here is to save your legs and start with the London Eye. See it all from the air and then spend more time at whichever one or two interest you most, rather than running around like a madman.

Shopping...or just browsing:
Forget Harrods unless you want to laugh at the prices. Visit Camden Market instead and see all the punks, mods, hippies, etc. It's a great place to pick up bargains too, and a melting pot of English style.