So it turns out National Express is not the only cheap way to transfer from the airport.
I found this website: http://www.gatwick-airport-guide.co.uk/transport.html#a
and they gave me a number of options. One of them is easybus.co.uk
They are cheaper than national express and they stop only at your destination.
They take you from Gatwick to two major stations in downtown London, and it is easy to find hotels nearby.