Personally, I would stay in London to maximize your two days of sightseeing, shopping etc....
The hotel rates are coming down at the moment.... Search for the Thistle Hotel which if I remember is near marble arch. Stayed there once and the rooms are Gulf size. Location is great and the price was not 150 pounds a night....

go online to find the gatwick express to double check the travel costs into London. Weigh out the time you will loose and put all that up against the extra cost of a hotel in London.

The B&B's.....always best to have one recommended by someone I believe.