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Does the light light up when you press the button? I you just not getting water but Ice still comes out of the door?
Okay well the click is from the main control board and if it's humming it sounds like it's getting power so yes I think the valve is bad, it's trying to open but can't you really would need to test it with a voltmeter to be sure it's getting steady power to the valve while the buttons pushed, chances I however sounds like the valve is gone, you can order the valve at this blue link below,
http://www.repairclinic.com/PartDetail/Water-Inlet-Valve/880070?modelNumber=GSL25KGRBBS thanks and well wishes, Bryan