You definitely HAVE to go to the Tate Modern. And if possible, you should see the Turners at the original Tate Gallery. (Turner is one of my favorite artists.) As for shows, it does look like you are narrowly missing some. But here are a few ideas:
10/9 Clinic, To Rococo Rot, etc. @
Hackney Empire 10/12 Bloc Party @
Lock 17 (in Camden!) -and- Club AC30 as Medusa recommended (If you can only make it to one of these, go to AC30.)
10/13 !!! @
The Astoria (very convenient location)
10/14 Seafood @ ULU (Don't miss Seafood!!)
There are some others (Razorlight), but they are sold out...
If you wanted to travel outside of London to Oxford, you could see several other bands like the 22-20s. And The Blueskins are playing a show in Nottingham.
I always try to catch at least one play while in London. You may want to check the theatre guide when you get there. I haven't looked at the listings lately because it's too painful to miss such good shows.
Perfect that you are staying near Camden. There is a lot going on in that area. The street markets alone are worth checking out. Oh! A terrific tiny venue is the
Notting Hill Arts Club . Just a hangout during the day, but at night they have shows.