Let's be honest most of London's attractions are do-able in the rain - they have to be.
Museums (V&A, Natural History, Science, British Museum, London Transport)
Art Galleries (National, National Portrait, Tate Britain, Tate Modern)
Shopping
Ripley's (in Leicester Square)
Madame Tussauds (I don't think the planetarium exists any more)
Cathedrals (St. Paul's, Westminster, Westminster Abbey)
London Zoo - large parts here are covered
Theatres
Resaurants
Covent Garden (shops, cafes, market stalls)