There is only a very small amount of accommodation available in central London. Most of it costs over £200pppn, with some rooms at over £500pppn. (If you consider that it costs over a million pounds to buy a one-bedroom flat in some parts of central London, it should be obvious as to why there aren't many people buying up big houses in those areas in order to turn them into budget hotels!)
Well over 90% of London's hotels are in areas such as Kings Cross and Bayswater (which are just a few minutes Tube journey from the heart of London), so that's where most tourists stay. (Some people also stay a little further out, in areas such as Stratford, from where it's also easy to get into the centre of London via the Central and Jubilee Lines).
So it might be better to look for somewhere where you can simply hop on a Tube service.
However this hotel is popular with ABers and, while certainly not 'cheap', isn't mega-expensive either:
https://www.strandpalacehotel.co.uk/
It's just a short walk from Trafalgar Square.
Otherwise you could look for accommodation in Bloomsbury (near to the British Museum) which is 'walkable' from Trafalgar Square, possibly using 'Russell Square' as your search location, as here:
http://www.tripadvisor.co.uk/HotelsNear-g186338-d189225-Russell_Square-London_England.html
Even so, I still think that you'd do better to search for budget accommodation which you can get to easily on the Tube. Use this website and look for 'Kings Cross', 'Islington', 'Victoria', 'Paddington' and 'Bayswater' in the addresses and descriptions:
http://www.hostelworld.com
(Don't be put off by the word 'hostel' in that URL. Simply carry out an initial search and then use the filters to select only 'Hotels' and 'B&Bs').