iMac Shuts Down and Won't Restart or won't start at all.

iMac is a 27", late 2013 model. It is new to me having bought it from a lady who never used it. I don't think she was selling a piece of junk and she has actually offered to help pay for repairs if required. When I purchased the computer it had been wiped clean as per Apple instructions. I brought it home and set it up with my profile info. At the time of purchase the OS was Yosemite which I upgraded to Sierra. From there I installed some programs that I had been using on my other iMac rather than migrating a whole lot of stuff over.

When I was done for the day I put the computer to sleep. Came back to it maybe 36 hours later and the computer was actually off. It would not restart at all. Totally dead. Used a different power cable with the same result. It is getting power as I can see a green LED on through the cooling slots. The only way that I have been able to get this thing to start is to unplug it from the wall and let it sit for about 4 hours. Then it may boot up on occasion but not always. On those occasions I have been able to boot I've used Disk Recovery and Disk Utility reports no problems. Biggest problem is that when I do manage to get it to start up it's short lived. Best I've got is 10 minutes of use and then the screen goes blank as if the plug were pulled. Sometimes when starting I'll get a quarter way up the progress bar and it will just go blank. Other times I only get the start up chime and then nothing more. I've been able to install a trial version of TG Pro and it shows all internal temperature to be normal.

I'm at a loss. I've contacted Apple support with no resolution. I have tried both resetting the SMC and the NVRAM with no improvement. I have an appointment at the Genius Bar early next week but don't see anything coming from that other than leaving my computer behind.