As strange (and scary) as it may sound--every time I go to London, I book only my first night & then I shop around the area I stay in for a cheaper hotel.
I have ALWAYS ended up finding less expensive rooms this way.
I normally stay just a tad east of Kensington, close to the tube, and then get around from there, but I don't know if that's central enough for you.
There are tons of walk ups/3-4 story homes that have converted to hotels and I've found them to be a.) reasonably priced, b.) excellent locations in relationship to tube, market & sites, c.) nice accommodations, but not as large as the hotel rooms would be in a hotel chain.
Here is the link to one of the places I've stayed:
http://www.bromhotel.com/rates.htm
** Right now, the room is 100, but when I went to look up the place for you, i saw that they were doing a feb/march special where, if you book online, they take 20 off of each night's rate.
If you don't like this one (or the area), you can do a google search and hundreds of the little hotels should pop up!
have fun!