I'm guessing a bit here, but I have a feeling that you are not on the latest version of the IDE (the firmware update shouldn't try a downgrade). I would suggest to try out the latest version from Galileo Software Downloads.
The version numbers was a bit weird to begin with (all called 1.5.3), but lately they have added an additional number at the end that increments, It's on 1.5.3-Intel.1.0.4 now and should also reflect that in the window titlebar.
In the Tools -> Boards menu, there will be a total of 3 different boards, "Intel Edison", "Intel Galileo" and "Intel Galileo Gen 2", and it's of course the last one you need to pick.
If I'm not mistaking, the onboard LED has it's own I/O, but it's hooked up in software by the IDE to blink with pin 13 (to be compatible with the regular Arduino).
I hope it helps, if not, we will need to dig in a bit deeper ;-)
Thank you flummer. We only loaded the software today, but somehow Google led us to a slightly out of date version. I was running 1.5.3 as you thought, and when I followed your link I was able to download the latest 1.5.3-Intel.1.0.4, and that worked straight away. Very much appreciated!