If the LEDs go dark on the router port or the device card port, that indicates a loss of power to the LAN feature of the computer. Nearly every machine I have seen in the past years shows an active link to the router just after the machine is turned on, but not yet activating the LAN links (networking).Don't know if this datum will make any difference but, with the computer turned off the router shows the 'Link/Traffic' LED on--this LED is on solid and flashes when there is traffic for this port. BUT, the same port's 'Speed' LED (on the router) is OFF--this LED is normally off for 10Mbps (or when no link), yellow for 100Mbps, and green for 1000Mbps, and it remains lit for the other computers when turned off; for this computer it is green when the computer is on.
If the machine truly shuts down then the PSU has stopped providing power to all circuits, except the PSU would still provide a +5 stand-by voltage to the USB ports, BIOS circuits and other select features as long as mains power enters the PSU.
If there is a jumper or BIOS feature to allow powering on from the keyboard, that may be a possible cause, if not enabled.