Those two airports are a long way from the city, and each other. Although I have been to the approximate midpoint of the two(for great Indian food), getting there from central London is a long trip. So, I would also advise to stay in the Bloomsbury/Kings Cross area, also. I prefer Bloomsbury as I can spend days enjoying walking around and do not have to use public transportation.

Consider using a car(hired) for the trips to/from LHR. The cost is around £35-45.

Hotels in London are quite high for what you get. For cheap, also look at http://216.65.16.70/cgi-bin/bookings/promolist.cgi It is the St. Giles hotel over The tottenheim court tube.

I have no idea how to get to/from LCY, but if you search this site, I know the question has been addressed.

Also look in London in the travel forum.