I know you're primarily interested in London, and who could blame you? It's a fabulous town.
Have you considered staying in Brighton? It's a beautiful seaside town, it isn't horribly far from London, and they have a decent selection of hostiles. It has a day life, a night life and often times concerts and festivals are held there in the summer, so you'd be in the middle of the action. It has its own train station in the middle of the city that you could get to by foot. I also don't think it'll be nearly as expensive as staying in London.
I just think if living in England is truly your passion, staying in London the entire time and being on "tourist mode" for a month isn't going to give you any insight to what it's like to live here.
I live two miles from Gatwick airport, and often see planes flying over our house. Nearby we have Arundel (gorgeous little town with a castle well worth seeing), Hever Castle, Hampton Court, Leeds Castle, various medieval summer festivals all within an hour's travel from our place.
Contemplate Brighton and maybe book a London hotel via priceline for a couple of long weekends. Then you'd probably get to see more of England that way.