Ok, first of all look for a HMV or Virgin Megastores shop. They usually sell older singles, and HMV is usually more reliable. Also be on the lookout for smaller record shops as the might have a second-hand singles section.
As for the sightseeing I recommend you visit some of the museums, the Planetarium is a very interesting place to visit, the Tower of London, you could look at Buckingham Palace from the outside, and there are so many more.
I know a very good beach as well if you are interested, it would take you about 100 minutes to reach there from London, but would be well worth it.
