You should be eligible for housing benefit, but because you're under 25, you'll only be able to get the rate for a shared room. Your best bet would probably be to try to find a room in a shared house as the deposit will be a lot lower than for a flat. Saying that, though, it's possible to get flats for £250 a month, but I suspect not around London. I rented a flat in a town called Dunstable for £395 a month. Sometimes a landlord will accept a 'deposit bond' from the council in lieu of you paying a lump sum which is really hard to find. The moving in costs for my flat in Dunstable came to like £900.
Have you tried the council? The waiting lists for council flats are very long, but if you're homeless (but not intentionally homeless - your dad needs to have kicked you out, and you need to have no other place to go), it's considered an emergency and they'll probably put you in a bed and breakfast for a few days until you can sort yourself out a place.
Hope things work out!