score:7
You need the Youth Hostel Association:
http://www.yha.org.uk/places-to-stay/london
Central London beds from Β£15/person/night - not quite as cheap as you asked for but probably one of the cheapest you're likely to get, short of a tent or a homeless shelter.
Actually, looking on Airbnb there are a couple of places offering beds for as low as Β£7, and a couple of them are in what I'd consider central London. Use the price filter to find them, and I suspect you'll need to book way ahead of time.
https://www.airbnb.co.uk/s/London--United-Kingdom?price_max=8
Update: I'm looking at Airbnb again later the same day and the central London places I saw offering beds for for Β£7 are no longer showing as listed - perhaps because they're all booked up? I don't know. There's still a few places showing at that price in outer London - but I guess the moral is that it's worth checking airbnb more than once because places go in and out of availability, even if you're not searching for specific dates.