Whilst there is no such thing as a cheap hotel in London I would definately recommend taking a look at the Travelodge hotel chain in London which aims to be the biggest hotel brand in Central London (within the North and South Circular), with a target of 7,000 rooms in the Capital in time for the London 2012 Olympics.
We stayed over earlier this year at the
Travelodge hotel on the London City Road, EC1Y 1AE in time for the Chinese New Year Celebrations, and saved a substantial amount of money, the hotel was very close to the underground so we were able to do some sightseeing in London on Saturday before the event.
The room was clean and tidy with a nice big double bed plus bath and shower. Nothing flash, but ideal for sightseeing purposes. We paid less than £50 per night for this double room which considering there was an event taking place in London I thought was very reasonable.
If you're interested, take a look at the
Travelodge website for an up-to-date list of hotels and special offers.