When it comes to London the words "cheap" and "accomodation" do not go together anymore than the words "quick" and "beer" do in the rest of the country.
There is a reason why several million Londoners commute into the City/West End on a daily basis and that reason is the non-existence of Cheap Accomodation.
The nearest you are going to get to 'cheap', (that being a reletive term), to Liverpool St (the most expensive office space in Europe), is to either head out to the East End, or the NE/E/SE Suburbs.
The other option is to join one of the clubs/societies that run central London hotels and hostels. These include the ex Forces, Scouts, YMCA, various Church based organisations, AA, RAC, etc etc