The best non-member YHA hostel price I found in London for a random date in May was $32.22. Member prices are a bit better - ~10%?. The YHA hostels are rated, and the highest rated London one had a low price of $44.83.
This would be a dorm bed with linen, and probably a light breakfast. Showers. Probably kitchen privileges.
You don't mention your age, but if you are more senior they can have a bit of sympathy and give you a bottom bunk.
It took me a while to figure this out, but grocery stores in central London are located in the basements of many of the big department stores.
Romelle