You will find London room prices ridiculous (though as I see you are from NY, maybe not as shocking to you as others). It is unlikely that you will get a room in anything other than very basic accommodation for less than £200 (£ not $) per night in a reasonable, safe area. I would suggest staying further out and taking day trips into town, rather than the other way around. Within an hour train travel of London you can be in some beautiful countryside from almost any direction, still with historic sites to see.

Remember that wherever you are in central London you will need to use buses and/or tubes and each journey is likely to be in the region of half an hour, especially some of the tubes.

A good way to be around London is to use the visitor buses. You pay for 24 hours and can get on and off as often as you wish, somewhere takes your fancy to see at closer quarters, just get off and enjoy the sights, get on a later bus. They cover just about everywhere you want to visit. Current costs are about £15 per person and tickets last 24 hours, so buy a ticket at 11am one day, get up early the next and you're still good til 11. If you do stay in London itself I would recommend staying away from Earls Court, Bayswater, Soho and most of the east.

Enjoy your trip to England, but unless you only want to see a large city, please remember that there is an awful lot more to England away from London, as there is US to NY.