Assuming you don't want to hire somewhere for your group alone, the Southwark Rooms might fit the bill, but the music tends more towards chart (pop, dance and indie) on the Friday nights that I've been there.
If there are a few of you, you'll probably want to reserve an area - and I've always found the staff very helpful at making sure you have drinks etc if you're in a large group.
No connection - and I should say the last time I was there was about nine months ago.
Edited to say, I've just looked at the new website for the rooms and the picture of the upstairs doesn't really do the place justice. There is a dancefloor area downstairs which is pretty relaxed and cosy. Worth a look.
there's a great place near Old st, i think it's called 'the bedroom bar', it's above or near a comedy cafe type place. The serve great mochittos' (sp? tequila thing), and it's the first place that i saw Desperado's being sold.
Try asking the Market Porter in Stoney Street, Borough Market if their upstairs room is available. I had my civil partnership 'do' there, the food provided was great, and the atmosphere of being around the market in the evening is great. People were dancing in the streets to the music my DJ was playing, it was a special night and the staff were great. We had about 80 people up their which was about the maximum it would take realistically.