The best thing to do is not stay in London,its very expensive,but depends on what your looking for-London is a good place to visit but not a good place to stay.I lived in a little village called Hollingbourne,Kent near Leeds Castle,its on the main train line to Victoria station so its very handy,and you have all the beautiful countryside around you and some reasonable places to stay,but on the other hand if you stay in London then keep away from some of the more untasteful areas,westminster itself is not too bad but a nice place to stay is Putney.To be honest London is a bit of a crap hole,the history is cool but thats about it,but like i say depends on what you want to do,let me know and i will point you in the right direction