I am so envious! My votes:
--evening ceremony of the keys is one of the most amazing events in London.
--If you go to the Cabinet War rooms at the end of the day, you may be the only one down there---really lets you mull over the events that occurred.
--If you take a boat ride on the Thames, be certain to find out how long the rides are, and when they return to your dock---a wonderful ride, but some of the journeys can take an entire day and I don't think you have that kind of time
--Completely agree with Kristilew re the 100 Objects; overtherwise, you can get sucked up into an entire day at the British Museum. My picks at the Museum: Elgen Marbles; Rosetta Stone; Lewis Chessmen
--A nice lunch and walk:
----Fordham and Mason
http://www.fortnumandmason.com/the-store.aspx
for treats to bring home, with lunch in the upstairs Fountain Restaurant for Welch Rarebit. I always do this my first morning in London---the rarebit and a glass of white wine revive me!
(
http://www.fortnumandmason.com/the_fountain_restaurant.aspx) ;
----then look around Picadilly Circle;
----then walk down to St. Martin in the Field (check to see if there is an afternoon rehearsal and slip in to hear it
----see the lions at Trafalgar square;
----walk past White Hall and the memorials;
----go past Big Ben and Parliament to Westminster Abbey, and go inside for evensong (it is free if you are a worshiper) and really take in the lovely music;
----back to hotel to freshen for dinner!
--I have taken many of the walking tours that others have referenced, and often at night. I am often in London alone, and these tours are a GREAT way to see the little nooks of London and meet interesting people.
--The British Library has reading rooms where you can request any number of rare and fine manuscripts, don white gloves, and view them yourself! Check the details on the website....
So much to do!