If you like metal, I recommend having a drink in the Dev in Camden, if you like art and world/arty films, check what's on at the ICA near Trafalgar Square, if you want a fun queer night out I recommend seeing what's at the Bethnal Green WMC or Soho Theatre, if you like experimental electronic music, see what's on at Iklectik, if you want a goth thing, there's Kaos, Monster Queen and Slimelight.. what else, wander round Epping Forest or Hampstead Heath, go to the big museums, they're all free (V&A, Natural History Museum, Science Museum, British Museum, TATE modern, TATE britain, National Gallery)
And if you like Italian pizza/pasta, La Porchetta in Soho I really like - might be worth going if you go for a look round Leicester Square/Piccadilly (just to see the area)
Also for shopping, that market down Brick Lane, where you can also get a good curry and visit the Whitechapel gallery nearby
Hackney wick area for graffiti art and warehouse parties, dunno how buzzing it will be now the weather is shittier though