I've booked this place:
http://arosfalondon.com
Bloomsbury district,
Arosfa Hotel
83 Gower Street
London
WC1E 6HJ

About $200 (£125) night. More than I expected to spend but I kind of gave up looking for cheaper. I'm going pretty cheap in France so it'll average out.

Thanks to everyone who chipped in with advice.

The Arosfa is a long established and well run budget hotel. Not at all posh but a very good option. Lucky you got in, it tends to book up pretty much year round.