Ah .. The other thing is that in the BIOS settings, one can select the PC to automatically start up, when a keyboard 'event' was recorded. You have to look through BIOS to find the 'wake on event' setting. Change it, save and exit and restart. Some PC blue screens or auto-reboot errors 'may' load the 'optimal default' or 'default' BIOS settings after these errors, so will need setting back again.
Is the mouse corded to the PS2 (round mouse) connector? And the keyboard to the keyboard connector?