Hostelling International, formerly known as International Youth Hostel, is a very progressive (and hip, young and all that too) company that runs hostels all over, might check them out. I know they have a few in London, depends where in the city you want to be.
Many of the major hotel chains, particularly the more spendy ones like 4 Seasons, W, Intercontinental and the like are pretty open and accommodating of anyone who's dropping that kind of money. Hotel people are a lot like emergency room types, they've seen it all, and then some. Every hotel has a little bit of the Overlook in it.