@VoytekZ
Perhaps you should ask this somebody who's more familiar with London than me because I'm staying here only for a few months
But according to my point of view, you
must see these things (well these are the best known landmarks and they DO kick ass regardless of how much hype they have caused and how popular they are

):
1. Houses of Parliament and surrounding stuff (Westminster Cathedral etc., some parks).
2. Trafalgar Square
3. Canary Wharf
4. Take a walk around London City (Swiss Re, London Tower etc.)
5. Take a walk around West End at night and go to some nice pubs or restaurants (it's extremely expensive but you just can get a few beers or cocktails what won't make you too poor anyway

)
6. Go to Piccadilly Circus and surrounding areas (it's the West End as well)
7. Take a night walk around the pedestrian riverbank near the Tower Bridge.
And the rest 3 just pick by yourself. Buy a day travelcard for the public transport and use public transport for the whole day to get from one place to another (it doesn't take that long). For £5.20 you'll be able to travel all around London using tube, buses and trains. So just select your point of interest and get your ass to it (just as I did during my first days in London

).
P.S. You can easily do this in 2 days (or even 1 day if you're fast and won't mind being extremely exhausted by the end of day)
