Use this website:
http://www.hostelworld.com
NB: Don't be put off by the word 'hostel' in the URL. Other types of accommodation are offered!
Carry out an initial search but then use the filtering options to select 'Apartments' (while deselecting everything else).
Among the cheaper places you'll find, 'Bed In London' (176 Commercial Road, Aldgate) looks to be an interesting option. (Aldgate Tube Station is on the Circle Line and offers quick access to the all of central London's tourist attractions.
However your post suggests that you might be travelling on your own. ('Bed In London' only offers accommodation for 2 or more people). If so, check out 'All Star Private Studios'. The reviews are good, and access to central London (via the Jubilee Line) is quick and easy.
Remember that you'll be travelling to one of the world's most expensive cities for visitors (and in the year which sees that city host the Olympic Games). You won't find anything really cheap.
Chris