Don't know if your electronically inclined but you can take the board out and try cleaning around the power switch contact or see if it's come loose. Check the power jack connector too. If you can't see anything loose, cold solder joints or dust buildup...a new unit or power supply may be your best bet.
I think your best bet would be to sell some of them individually like maybe River Raid, Plaque Attack, Oink and Crackpots assuming they are in mint condition and maybe have manuals. The rest, bunch together in a lot. That's what I would do.