My own list would be like:
The British Museum
Victoria and Albert Museum
see a play – Macbeth at The Globe sounds great.
walk through a big park, e.g. Hyde Park
eat some British food – e.g. see how good the fish & chips is/are
At least take a peek at, if not necessarily tour (or do, if interested):
The Thames River and Tower Bridge
St Paul’s Cathedral
The Tower of London (also good inside, but not must-see for me after seeing it when young)
Big Ben & Parliament
Buckingham Palace
Westminster Abbey
Trafalgar Square
Harrod’s
ride the underground “Tube”
Also good are the MANY other excellent museums, e.g.:
the Tate museums
National Gallery & National Portrait Gallery
Science Musuem
the Wallace Collection (if you are into medieval arms & armor)
London Transport Museum
London Zoo
Natural History Museum
If you have time to get slightly further out (easy on train from Tube):
Hampton Court (country palace with hedge maze)
Kew gardens (nice botanical gardens with tropical greenhouses etc)