I say it every time someone asks but
Gordon's Wine Bar is one of my favourite places in London to sit and have a drink (the building in which it is located was one inhabited by Samuel Pepys and later by Rudyard Kiping).
I would also highly recommend visiting
Borough Market and check out the farmer's market. Also visit Neal's Yard Dairy, which is located there. One of the best cheese shops I've ever been to.
The Tate Modern and the Tate are both excellent galleries.
I second a visit to the eye, but if you don't want to spend the money you can also climb to the top of St. Paul's Cathedral.
If you've read Neal Stephenson's Baroque Cycle, there are all sorts of places along the Strand, Threadneedle, etc. that are kind of cool to visit just because they are so prominent in the books.
London is full of little surprises and I find that the best way to tackle it is on foot.