^ A fruitful trip, I see.

It's tragic, but so funny. Is that for
real?
I would highly recommend a boat ride from the Thames Southbank if you do decide to go to Greenwich - the greenwich observatory, maritime museum, and surrounding views are really worth it (they also have a planetarium at the observatory which has some great short movies - the "blackholes" showing rocks!). Plus, from Embankment station, which is directly on the water, a boat ride from there would only be 45 mins max to Greenwich. The Thames Clipper is the fastest way to get there, and costs about £4 each way. During your ride, you can have uninterrupted views of all the major landmarks - big ben and the houses of parliament, St Paul's cathedral, canary wharf and the "gherkin" building, etc... It's quite a refreshing ride on a hot summers day.
http://www.thamesclippers.com/
If you are interested in museums and art, I highly recommend the Courtauld gallery, which has quite a surprising number of masterpieces:
http://www.courtauld.ac.uk/gallery/collections/index.shtml
Just as impressive, if not more so, than the National Gallery, but less swarms of people, and the location itself - Somerset House, is breathtaking. It has a wonderful quiet, regal, gravitas to it. Lots of paintings from the Renaissance, and a whole room full of Rubens, if you're into that. This one is not free, but it's only about £5 pounds per person.
Buckingham Palace, you have to see, if only for St James' Park alone - which has sun loungers liberally scattered on warm days. So pristine and peaceful, and a nice place to rest after Piccadilly and Trafalgar square, which are very close by. Plenty of wildlife there, a variety of birds, lots of squirrels, also black swans in the lake along with the white swans - I don't think I've seen black swans anywhere else. I'm sure there must be, but those took me by surprise. There's a guy who likes to sun himself there with his python, another one who likes to take his ferrets for a walk along the paths on leashes... The human-life might prove just as entertaining as the wildlife...
If you enjoy a good drink - the best (properly equipped) bar in town is Lab, on Old Compton street - they have any and all ingredients, and will make any drink you describe, say by mentioning your favourite flavours - they'll whip up an especially designed one on the spot... the bar staff can even toss a few bottles and shakers ala Cocktail.

It has a similar reputation to Harry's Bar in Venice, in terms of excellence. It doesn't seem to have that many tourists though - more of a drinking establishment for the local media houses. It's not cheap, but it's not exorbitant either - cocktails average at about £7 each.
http://www.urbanpath.com/london/cocktail-bars/lab.htm#main
I could go on and on, but 5 days is not enough to get a proper taste of London, you would need at least a month for that. So you'll just have to prioritise according to preference. What a wonderful problem to have.

Perhaps if you are specific about any particular interests, you might get more tailor-made suggestions...
...And my signature is
on topic for once!
EDIT: to provide links.