If you can get out of London at all what about a visit to Stratford to RSC, that's got to be worth a little visit!
As for things to do in good old London Town, 'The Sound of Music' in the West-End at the moment is quite spectacular if you like Musical Theatre. If your looking to get away from the Theatre thing then just doing the normal sites like Trafalgar Square, The South Bank (where you'll find the National Theatre, The London Eye, The Tate Modern), Buckingham Palace etc. etc can be fun! If your looking for nice bars, pubs, resteraunts and shopping go to Angel which is a lovely area! If your looking for a fun (/ very camp!) night out hit Soho where there's plenty of fun to be had.
Good (torusity) shopping for clothes and electronics (which obviously you'll find so much more expensive over here!) can be found at Oxford Street.
If you've got any more specific questions just ask and I'll try my best!
Sam x