You will be be in Piccadily so a few things to do:
As someone else said, go to Fortnum and Mason nearby. Shop for the VERY rich. Amazing food hall, and other amazing floors.
http://www.fortnumandmason.com/
There is also St James Church in Piccadily. Designed by Wren who also did St Pauls. Much smaller of course, but lovely inside (cool as well)
http://www.st-james-piccadilly.org/
Burlington Arcade is also nearby, continuing the theme of the rich. Amazing small arcade
http://www.burlington-arcade.co.uk/home.htm
And to finish the theme of the rich, Bond Street with all its jewellers and designer shops is also nearby
http://www.streetsensation.co.uk/mayfair/bs_in tro.htm
Go there and count the Rolls Royce cars.
All the places I have mentioned are just a few yards from the Ritz.