My one recommendation for you would have been the British Museum, but as you don't want museums or galleries I'll suggest the British Library instead. Right next to King's Cross and St. Pancras (which are lovely buildings in their own right), it's got some wonderful collections and exhibitions, including (my personal favourite) the Ritblat Gallery.

Just plan a walk around, see what you can see. Some definite recommendations are:

• Covent Garden
• Trafalgar Square
• The walk from Trafalgar Square to Westminster, along Whitehall
• Buckingham Palace and Hyde Park
• Carnaby Street, if you like your shopping
• Wandering east from Carnaby through Soho
• St. Paul's cathedral, and crossing the bridge to the South Bank by Tate Modern
• Going further afield to Greenwich.

Let me know if you want more specifics, I can throw together a rough google map with a suggested walking path if you like.