London is fantastic for the solo traveller.
If you like art and museums then head to:
Victoria and Albert
Natural History Museum
Also go on a walking tour - download one from a website or join a professional one - I love going to the City on a Saturday morning when its all quiet
Visit some of the churches such as St Pauls or Wesminster Abbey
I adore the Tower of London and love mooching along the South Bank
If you are happy to get out and about a bit, then head to Kew Gardens, the RAF Museum, Windsor, Greenwich etc. almost 6 years ago