The one I used serves a number of postcodes in East London (and had another location in Essex) and was called "The Ironing Board". They charge by weight, their site is here, but you have to call them rather than arrange it via the web - http://www.theironingboard.com/

I would expect there to be others in other parts of the city. I could be wrong :)

--edit-- It looks like they've been doing it for 21 years!