When I press the power button, my computer won't turn on. No light, no fan, no beeps, nothing.

1. The outlet being used has power. I've tested the monitor in the outlet being used and it works. I've also tested the surge protector and it is functional.

2. There are no power buttons or switches in the back of the computer or monitor or on the power supply, and the surge protector IS turned on.

3. I tested another power cable that worked on another computer, so it's not the cable.

4. All cables are securely fastened.

5. All internal components and cables were reseated.

Any other ideas?

If the PSU (Power Supply Unit) fan doesn't start turning then I would suspect it has died, or else the power button wiring has become disconnected.

If everything is connected snugly then you probably need a new PSU - these do fail, particularly the cheap ones that are often supplied with pre-built systems.

You can get a PSU tester here
(Thanks to Bill_Bright for that link)