I'm not sure how the National Gallery isn't a 'tourist landmark'...I'd recommend the Tate Modern every time, no question.
If you like art, I'd also check out White Cube gallery, and Saatchi gallery. The Royal Academy of Art usually has some interesting shows too.
If you like quirky museums, check out the Old St. Thomas's Operating Theatre, which is quite a strange visit. Or the Sherlock Holmes Museum.
There are lots of beautiful gardens in London, so I'd suggest a visit to Kensington gardens or Holland Park (which has a Japanese themed garden contained inside it).
London Markets are also a fun way to spend an afternoon, Portobello Market is one of the best known.
Prince Charles Cinema near Leicester square runs old films for the cheapest prices you can find in London, pretty much.
Hope that gives you some ideas
