I live in North England and love it and I also love London.
I recommend you get yourself to one of London's famous clubs to sample some of this world's best house music, London style. You will never forget that experience! I recommend The End or Fabric or just ask someone when you get there.
During the day a walk around Campden market is always exciting. Get the Northern line tube and get off at Campden (funnily enough) and just wander.
There's just sooooooo much to see and do in London.
I lived in London for a while after finishing uni and have visited more times than I can remember. I love London, theres so much to see and do. What i like is that you can just walk around (provided you have some sort of map or know roughly where you are) and you can walk for miles and just hop on the tube to take you back home.
BUT I feel i must get this in - London is only one small part of the UK!! there are so many other places worth visiting. I worked with a girl in London from Australia and she lived in Edinburgh for a couple of months and said this was her favourite city from her whole trip! I am obviously going to say visit Scotland as im from there but there really are some amazing places to see. Also remember you will probably spend less money if you dont spend all your time in London. The UK is small compared to places like Australia and the US etc so get out there and see as much as you can. There are loads of amazing cities, Manchester, Liverpool, Glasgow(where im from!!) are all def worth a visit. and if you book intime you can get fairly cheap train tickets to these places - or even cheaper bus tickets.
enjoy your trip!