GFCI Won't rest. shows no LED Light but has power
GFCI receptacles very rarely go bad. So before your run out, buy and replace a GFCI. First let's try and determine if it is bad GFCI or something else is wrong. First unplug every appliance and turn off every light on the circuit and see if the GFCI will reset. If it does reset then try plugging in and run every appliance and turn on every light one at a time. If any appliance trips the GFCI then your appliance is bad and it needs to be replaced. If turning your light on and the GFCI trips then the fixture is probably malfunctioning and needs to be repaired.

If the GFCI trips when no load is on it, then after you turn your power off, pull you receptacle out and disconnect the load side conductors of on the GFCI. Carefully turn your power on and see if your GFCI resets. If it does not reset then it may be bad, but if it does reset then you have a problem with you downstream wiring. Now you have to go through the entire circuit and check to see what is in failure. 

That means you reconnect the load side conductors and remove each device in sequence and test to see when you get a trip. Once you get a trip you have isolated the problem and repairs need to be made.

It's really a step by step procedure, but it is also why when you call an electrical contractor they can't give you a set price since they don't know how long it will take find and repair the problem.

Hopefully it will just be a bad appliance or GFCI. Good luck.