I was under the impression that the power LED should come on and stay on when the machine is powered up? FWIW I tried connecting the power supply from my ][ - which is known good - and got the same result. I have not tried connecting the //e's PSU to the ][, but it doesn't seem necessary. I think it's a safe guess that there's something wrong with the //e motherboard or I'm doing something wrong.
Man... I don't know, to my untrained eye, the entire board looks OK. My concern is that I've got an IC socket or several of them that are corroded to the point of not making contact with the chips in them. That in combination with legs rusted off and soldered back on... I don't know. Kinda feel like I am just shooting in the dark with so many possibilities.
The one solder joint that bothered me was one of the four caps near the power supply connector. I thought I had accidentally flowed solder over to a pad very close to it but on closer inspection it appears that it's supposed to be that way - it's on the same trace as one of the resistors in that area, or at least it appears to be.