I do not understand what you mean with the screen lights up 1 led .. could you explain that a little more to me?
My iMac from 2009 broke down, boom, turned off. I couldn't get a peep out of it, no lights turned on, no buzz, no hard drive spinning, it was indeed the power supply unit, apple fixed it for 140 euro or something.
But if you have an LED light up, it is getting power .. so I am curious to hear what you mean with that specifically.