I stayed in Travelodges in London, five days a week, for over a year. Central London, unless it's really out of season or you book mega early, you'll struggle to get one for less than £60/£70 a night. That adds up. The cheapest one was always Docklands, out on the DLR, I sometimes got that for £40 a night. It's a hole, and it's out of the way for most CP work, but it's there.
As someone has already mentioned, Earls Court always has something on a budget, I stayed there whilst working a job in Mayfair. £30 a night, 10 mins to Hyde Park Corner on the Piccadilly line, a pretty good option.
I you know you'll be down there for a stretch, there's one-month short term lets on "modern apartments" (bedsits) in Earls Court, Kensington, Baywater, Notting Hill Etc. You can get them for £250 a week, may sound a lot but that's £35 a night plus you get a cooker, fridge, washing machine etc which will save on expenses. Only worthwhile if you have a stretch of work squared away.