Lots of people recommend Stratford but I'd only recommend it if you're a massive Shakespeare fan. You can get the same experience in many other towns in the country. Stratford is very small and very, very boring. Everything closes early and it's really just not that special. However, if you do decide to come up the West Midlands way, Warwick Castle is always worth checking out.
In London, the places I 100% recommend are:
The London Dungeons
Tower of London
National Gallery (in Trafalgar Square - much, much better than the two Tates)
Hamley's on Regent Street
My favourite pub is The Coal Hole on the Strand, and there is another really nice one in Covent Garden called Porterhouse
Wandering around Covent Garden on an evening is really something.
Madame Tussauds is ok, but it's overpriced and it will be full of Japanese tourists who don't understand the concept of queuing.