the 4 pins provide additional power to the board rather than try to suck all the current through the 20 pins.
It may or may not solve the problem.
I would start looking to friends or open support locally.. see if anyone can allow you to borrow their power supply for a few minutes.. usually you don't have to pull the whole power supply out from one case, just get it close enough and run the cords if you can..
but also.. when you turn it on, do any other lights come on solid? Like the IDE light which might come on for a second but should blink off more than on during a system boot ... of course if you disconnected all other cables (ide/sata/floppy) this would not have stopped it booting if one of those devices were holding the system from boot ...
the spare parts.. does that include cpu and memory and mainboard ;) ? or were some of them purchased new?
It would come back to probably trial and error part swaping.. power supply might do.. but this could also be board, and cpu related.. so.. good luck finding parts from friends or a shop locally that will work with you to solve it ... not having additional working parts makes it hard to go further.