Do you actually get any cold air out the vents?
It may just be a bulb burned out behind the button.
If the fan comes on when activated that does imply the system is switching on, and is fairly easy to check on the 20v turbo as the compressor is on the front at the top of the engine bay. Just turn it on and see if the centre of the compressor is turning with the pulley.
If it is but the aircon is still not cold then all indications point to it being low on gas