I can't help you with finding somewhere in london, but as for the karaoke in a box idea. . . My partner and I went to Cardiff one weekend, and could only find private rooms. Considering we live in a flat which looks like a box, surrounded by karaoke equipment. . . we really thought it would be a waste of money. But being nosey went to investigate. . . acting really really thick but enthusiastic about the whole thing, the barman gave us a half an hour free trial.
Half an hour was enough for us: we lost no time what so ever choosing songs or working out the equipment, and with only two of us, we each sang a two solo songs and three duets. If it was a completely new experience and if their was more of us, we would have wanted longer.
It was soooooo much fun.
We later found a karaoke, with a poor system, no books and no patter. . . but an audience. . .
Still not sure which I prefered.
If you get the chance to give "karaoke in a box" a go, with a fellow singer, do so.
If you fancy travelling 215 miles come and see me.