Why won’t the computer enter the boot sequence — even though the power is working

Following a storm last night — though without lightning or thunder — my desktop won't start up.

Symptoms:

• The external power lights and one internal light on the motherboard light up. The three fans spin.
• The front power on/off button and the reboot button do not work.
• The power button which is part of the power supply unit does work.
• The CD player does not eject when the button is pressed, even when power-lights show that the computer is on. However, when the CD's ATA control cable is unplugged, the CD player does eject.
• None of the beeps indicating failure in the boot process.
• Nothing at all shows on the monitor, not even the BIOS or other introductory screens.
• The monitor works, when tested on other computers; I also tested this computer with another monitor, which also shows blank.
• To check if it is booting up but incapable of video input, I listened for the Ubuntu startup sound — nothing. I also tried accessing it over Samba and ping — nothing.
• So, the power supply unit seems OK. The boot sequence does not get as far as the harddisks or the CD.
• Edit based on answers: A voltmeter shows that the power supply unit outputs 5V and 12V as it should
  What should I now do to diagnose or repair it?

Answer following thread with Chris Harrod: The problem is in the motherboard.

Outcome: A professional PC service shop vigorously cleaned the motherboard and replaced the motherboard battery, which, the technician said, was at the heart of the problem.

That did the trick and the computer is fine.