The London School of Economics has a program for their students who get to 'manage' an apartment building. They rent to tourists. They also rent spare dorm rooms at various locations around London - most very central and near good tube and / or bus connections. Check out their website.
[www.lsevacations.co.uk/][1]
Be aware that it has problems with some browsers- so if it shows no vacancies, that is not necessarily the case. Contact them by email or by phone to be sure.
You could also check out Citadines. They run 'aparthotels'. Similar to apartments, but with some of the services of a hotel. They also have multiple central London locations. We stayed once at their Holborn/Covent Garden location when we were taking our kids. It had a single bedroom, a pull out sofa bed in the living room, a TV, nice sized bath room, and a good sized kitchenette. We saved money by buying snacks and breakfast stuff and eating in the apartment. You could even pack lunches and cook dinners if you wanted. That's how apartments (or hotel rooms with kitchenettes) save you money in London. Here's their website.
[http://www.citadines.com/en/index.html?gclid=CLrUx7-YjsoCFdEYHwodB_sH7Q][2]