You'll be able to manage no problem, London is so diverse I'm sure you'll find you fit right in!
I don't live in London, but I go there regularly and can probably recommend a few places. Shopping is fantastic on Oxford Street, the huge Topshop right out of Oxford Circus tube station is a must, bottom floor is amazing. Also Harrods, in Knightsbridge just for the spectacle that it is, you have to see it. The Camden Markets are also cool, some nice little pubs and hangouts in Camden too. There is also a massive new Westfield shopping centre opened up in White City, Shepherd's Bush. Haven't been yet myself as the launch was only a few weeks ago, might be worth checking out if you have time.
For the tourist type things I've been on the London eye, it was alright. Not my kind of thing but it's nice to say I've been on it atleast. London aquarium which is really close by I thought was a real let down. Houses of Parliament tour was interesting, mega security though, only available in Summer however. Madame Tussauds always fantastic silly fun and the London Dungeons if you can stomach it. There is also loads of bus and boat tours around London and on the Thames. Go for the Ducktours one! Ha ha! Other places that are just nice to walk around Hyde Park, Trafalgar etc.
You might even like to pop down to Broadway catch a musical or something, might be a little treat for you and your friend. Nice night out aswell.
The trendy, but what I'd class more normal nightclubs, where you would meet cool people (not like Chinawhite or anything!) are ones like Ministry of Sound, Fabric and Turnmills in Clerkenwell. Shoreditch's Plastic People it's kind of eclectic though and like Cargo Club and Herbal close by. The Lab on Old Compton Street, nice atmosphere, kind of packed though. Soho Revue Bar (might like to check events though unless you can handle a drag queens night!). The Vauxhall Renaissance Rooms Rollerdisco is something different to try. I also really like going to music evenings, kind of helps when you know the bars owner too! My favourite The Bedford in Balham. Good food, nice for relaxing. Also like Ronnie Scott's, also Soho. Nice jazz club, starts off a night for me. I'm thinking with all this information I have I might be going out too much though.
Anyway I wish you a fantastic trip!