Assuming you are not relocating to London permanently - won't they pay?!
Very few hotels in the City (count them on Beadle's hands). The nearest is Club Quarters on Ludgate Hill. Imagine this is priced for business custom though, as will almost everywhere else. Aside from Kings Cross dosshouses, there is little moderately priced hotel accommodation in London.
You could try Barbican YMCA or City YHA... Private rooms, Red Indian headdress optional