|"I wouldn't think there's anything wrong with the power supply, but I could be wrong"|
Power supplies convert AC voltage into several different DC voltage rails. The main three rails are +3.3v, +5v & +12v. It's possible that one (or two) of these rails has failed, but there's still enough power to light a few LEDs or spin a few fans.
eMachines is notorious for using low quality Bestec power supplies. Your problem *may* be caused by a bad CPU, but I'd be more suspicious of the power supply. Unfortunately, they're difficult to test. A multimeter or "power supply tester" can check for proper voltages, but there's no easy way to check for amperage. For example, the multimeter can confirm whether the +12v rail is within tolerance, but there's no way of knowing whether it can deliver the 12v at 18A or has become so old/weak that it can only deliver 12v at 2A.
Test the power supply or try another one.