Make sure you visit the Tower of London (free to walk around outside...) At the moment they have the most amazing display of poppies. I'll try and get a photo up later, we visited there a couple of weeks ago, and I was very impressed with the display. They are planting a ceramic poppy for every person who lost there life during the wars, approx 882000.
St Paul's is also a great visit, £16 entrance fee. Approx 500 steps to the top! No elevators...
'The Shard' is also quite impressive, but the lack of facilities at the top of the viewing platform doesn't warrant the £25 fee in my opinion.