Just unplugging your router frequently would do no harm .... unless you observed sparks .... This should not be a concern ....
The critical damage must have been done when there was an electrical storm .... or maybe somebody ran into a power pole transformer near your location .... Maybe when you lost power to your house, the circuits within the router were 'weakened' ...and subject to 'reduced life'.
....even if you replace the ZD, there may be other defective parts that would prevent repair of device .... Inspect the individual parts ... resistors, capacitors, and carefully clean up the burned part on the board ....
If anything else is defective, you need to find it.
That said, it would be interesting just to replace just the ZD itself, as a test, and see if the unit powered up ......
You will have to go to a local electronics parts store to get the ZD part ...
.... I would try 5V @ maybe 1 watt ... ....no, I'm not sure, but 5V won't hurt anything.
You could order the part from an internet parts house, but you would be billed for shipping .... to excess....
One other item to consider ...You need to obtain a voltmeter, and measure the DC voltage to ground at the original ZD part location on the circuit board.
This is a measurement with the unit plugged into the power adapter ...
If you do not see any voltage on the meter, then there is no point in buying a replacement zener diode.... Something else has in fact been damaged ....and you will have to fix that, whatever it is, before you install the ZD.