To live or for vacation?
I have been to Europe a couple of times and I always stay in youth hostels! I don't know how you feel about that, certain people don't like them. I love them, because they are DIRT cheap, and they provide you with the basic necessities, a bed, a community kitchen if you need it, bathrooms, and a shower. When I travel I want to really see the country, not just some over priced hotel in another country, I could see that here! If you are traveling, and not vacationing (I think there is a difference, traveling is more seeing a place, going out and exploring and actually seeing all the different sights of a place, and vacationing to me is just kind of being comfortable and doing whatever I want!) then these are awesome. When me and my friends went to Europe last summer we would go out that morning, and not come back until late that night, being on our feet all day. So I didn't see a point in getting a fancy hotel, so we basically needed a bed and that was it! Some people don't like the idea of sharing a room with 10 other people though, people you don't even know. Or sharing a bathroom/having to wait to get a shower, etc. I personally don't care, but I know some people just don't like this. Which is fine too
Some website for London hostels are:
If you don't want to stay at a youth hostel, some bed and breakfasts can also be really cheap.
If you are looking for a place to live, then please disregard all the above information and sorry I can't help you
I hope you find what you are looking for and if you need more help feel free to ask!