We recently ate at the Gilbert Scott at St P. The food was excellent, if a little pricey.
If you're up for something unusual, I'd quite fancy a walk to the top of the O2 -
http://www.theo2.co.uk/upattheo2
Not far from there, the new cable car ride over the Thames looks fun, and it's not expensive at about £4 each way -
http://www.tfl.gov.uk/modalpages/23828.aspx
The recently renovated Cutty Sark in Greenwich was a really enjoyable visit that I would recommend -
http://www.rmg.co.uk/cuttysark/
One of our favourite places to eat is the Oxo Tower, with great views over London -
http://www.harveynichols.com/oxo-tower-london, combined with a walk along the Embankment, and a visit to the Tate Modern (free entry for all)
I guess other items on your must see would be:
Buck Palace
The Tower of London
Westminster (Palaces, Bridge and Big Ben)
The London Eye (called something like EDF eye or similar if you Google)
My advice would be to plan all your days in advance, and don't try to cram too much in - it's surprising how much time it takes to travel from one place in the city to another!
I'm sure someone who lives locally will be along with more advice soon
Enjoy!