Num Lock key is not responding - ASUS laptop, Win 8.1

Num lock turns itself off, and the num lock key will not work to turn it on. Solutions that have *not* helped so far:

The registry fix (changing the value from 2147483648 to 2)

Disabling fast startup (was already disabled)

Disabling mouse keys (was already disabled)

I have not tried the BIOS fix, but as that is only to turn num lock on or off at startup, I have no faith that will work, because the few times I have been able to get the num pad going, the num lock has turned itself off after a very short time.

Moreover, when I select "turn on toggle keys" in the ease of access menu, I get the sound I'm supposed to be getting from caps lock, but nothing from num lock. My laptop does not have a light indicator for the num lock, by the way.

I did manage to turn on num lock briefly, though for the life of me I cannot remember how, and it stayed on for 20-30 seconds.

Also, and this is the really curious thing, it's only the 7 and 9 that won't work. They have alternate functions as "home" and "pgup" - and these functions do not work either. The 7 and 9 keys have, per this moment, no function whatsoever. The 1 and 3 have "end" and "pgdn" as alternate functions - these keys work for writing numbers, but only that. 2, 4, 5, 6 and 8 also write just fine.

Testing to see if 7 and 9 were defective, I plugged in an old Targus external keypad, which again worked for about 20-30 seconds, but then would not respond. I think it stopped responding when I left-clicked with the mouse in an old DOS game I was playing - it certainly coincided with that.

Probably related: the "home" and "pgup" keys above the num pad do not work, either. Their adjacent keys, "pgdn" and "end", work just fine.

I find it particularly interesting - and frustrating - that my laptop won't even accept an external solution. But what can be causing this?