https://atariage.com...Switch_Low.html (note this is 4 port version, 95% same with 2 port version)
May need to trace the power. Check the power switch IC 4013 which should be near the switch button. Pin 7 and 14 should be showing around 11-12V DC. If not, the problem is between the power in jack the the IC including a few resistors and couple transistors. Shorted transistor or burned resistor would prevent power from getting through.
If you are getting power at the IC, check pin 1 and 2 of the 7805 regulator (both of them), If there's no power, and you are getting "HIGH" signal from pin 13 of 4013 IC, one of the transistor isn't turning on or in one of the regulator, a dirty cart slot not making contact between 2 pins. If the 4013 is stuck LOW, either 4013 is bad or the switch is not working at all.
If you are getting power all the way to both 7805 regulators, check that you are getting 5v out on pin 3. If it's 0, replace the regulator. If it's 5v, pray you don't have a dead CPU or other big important chip.