AirBnB is going to be the best compromise between price & comfort.
If you are going for pure budget, try the couchsurfing London facebook groups etc, and see if anyone will let you use a sofa.
I've done the hostel thing a number of times. I wouldn't recommend it unless you plan to sleep in your own sleeping bag on top of the bed because otherwise you expose yourself to moderate risk of getting a skin parasite like scabies.