Macbook keyboard backlight doesn't work
For all kind of hardware-related issues, I will recommend that following,

Reset your SMC, NVRAM. You can find guides for SMC here, NVRAM here.

If that doesn't help, boot into Safe Mode, you can the guide here.

The last step, is of course, to Time Machine it and reinstall the OS. Reinstall, not upgrade.

The very last thing is Genius, or repair it.

Update: some other fixes

I forgot the following things to do before giving up.

a) Create a new user account and see if it works

b) Clear your cache with things like CleanMyMac or other free alternatives.