peckham's 9 minutes from LB by train.
1k a month? I'd live in London Bridge.
Yup, Hackney is decent for getting to the City/London Bridge.Stoke Newington, Dalston, Mare Street, and Hackney Central are all on direct bus routes to London Bridge too
Kennington.
15 mins on a bus from London Bridge, you've got plenty of night bus options too if you ever work night shifts.
You've also got the option of the tube, Northern Line only a few stops.

have we ruled out overland trains as a way in?
Greenwich is two stops out from London Bridge and you may be able to get somethin for £1000pm
except it's not really a place so much as the space between other places. It has nice bits though

What do you mean?
It has a small high street with useful shops, a few decent traditional pubs, a few style bars (if that's your sort of thing), quite a few restaurants, a super market, a park & quite a lot of nice streets.
Kennington is a pretty decent area if you ask me, I don't live there, but go there regularly as I have friends that live there.
And a theatre (isn't the Oval theatre in Kennington?)

What do you mean?
It has a small high street with useful shops, a few decent traditional pubs, a few style bars (if that's your sort of thing), quite a few restaurants, a super market, a park & quite a lot of nice streets.
Kennington is a pretty decent area if you ask me, I don't live there, but go there regularly as I have friends that live there.
http://www.gumtree.com/london/22/37147622.html (just an example)
Walking distance from London Bridge
Bloody hell that's right where I used to live in the 1980's!
The East London line is not functional at the moment which may make travelling from Wapping tricky.