Check for continuity on the wire side that is soldered to the power supply pins, if everything checks out follow the trace to where the voltage rectifier is. There are usually 4 pins to the rectifier,you will see on the rectifier + sign a -sign and 2 sine waves. The 2 sign waves are the AC voltage from the outlet. Set your meter on AC and check for outlet voltage, then switch to DC on your meter and connect your leeds to the + and - this will give you DC output. If nothing comes comes out of the rectifier then desolder it and bench test the pin diodes. If just 1 diode junction conducts both ways then replace the rectifier.
Another factor is that you have to consider is that if it's a switching power supply. Switching power supplies behave different because they use an oscillator to trigger the switching action. Depending on the power supply orientation, if there is a small transformer with multiple output pins then it's a clear indication of a switching system. The oscillator can be triggered either from an SCR, or a 4 pin photo-coupler (opto-isolator) which is located near the transformer.
Keep in mind that some power supplies have a 3 pin regulator on a heat sink after the rectifier.
When you press the power on button, the relay is engaged by a transistor driver, it's either an NPN or PNP package type. The transistor is located near the relay, just follow the trace from the coil side of the relay to the transistor, then set your meter in diode mode and test it. If the driver is faulty then replace the transistor.
post back your findings.
" What were dealing with is the battle between the future and the past, between the powers that were and the powers that have yet to be. " --- John Perry Barlow