I lived in London for a few years. Here are some of my favorite things to do.
-DRINK anywhere and everywhere. My favorite spot is on halloway rd. punk rocker bar called BIG RED. all the bartenders look like pin up girls and all the customers are covered in tattoos. drinks are cheap and room temp. enjoy! The Crow Bar in Soho is also good, but very tiny.
-LIVE MUSIC in some amazing clubs around the city. Old Blue Last, The Barfly, Round House, Academy, etc etc etc. Maybe check out the website TIME OUT to see whats popping in the night life. But don't be surprised if all this shit going down in Paris has put a damper on the nightlife.
-CULTURE is cool too. Check out the many free museums and art galleries. Some of my favorites are The Imperial War Museum, The Tate (modern or brit), and if its still there The Saachi Gallery is one of the coolest things I have ever seen in my life. But I heard its now on display in Britton Beach.
(all the most beautiful woman I have ever seen in the world were in London museums.)
other than that its basic tourist stuff. Take pics and get tattoos and buy shit.