Machine won't boot unless it has been plugged in for a day or so

The most likely cause of things like this is a bad/faulty power supply unit, however, as your machine uses non standard parts, you may have problems trying to find a replacement.

I'm not sure if this is the answer , but I did have a monitor which had bad capacitors .This particular monitor took maybe 1 day to start . Maybe your PSU's caps don't hold (enough)power and need a long time to get enough power to start ?