That's the third one now.
I have read sometimes they can die due to firmware corruption and can be restored by reflashing. I've done it (out of desperation; it in fact turned out to be a different problem), so I know the complete firmware image is out there, somewhere. It's also on my computer, somewhere; if you want it I can probably find it. FWIW, I have several PK3 and all are working fine.... other than the Sure Electronics clone which never worked.
I want to say: hold the button down while plugging it into the USB port and with the software open. It might do something.
I also want to say, if you are programming in assembly, there is no use of a debugger. The PK3 has pretty much zero advantage over the PK2 unless you need PK3 for device compatability or high speed debugger. So... if i were you, I'd also have/obtain a PK2. Sometimes it's just as good or better. It can save you countless hours of waiting and rebooting and clicking on dialog boxes. Some people have zero issues with the PK3, but if you're on your third one, you are not one of them.

The only advantage of using PK3 for programming a 10F in assembly would be if you're using MPLAB X and want the thing to automatically program when you recompile/assemble. So you will end up with a couple extra mouseclicks or keyboard strokes, in that case. (Someone may point out you can setup PK2 software to automatically program when it detects a change in HEX; but IME this is buggy. It works fine until your code fails to compile/assemble... in which case you have to set it up, again. And when writing new code, you are going to have that happen pretty much nonstop.)
Well, for the really old baseline stuff, which your PIC is not, the PK3 can also program faster because the PK2 redundantly writes all empty code due to stupid way it handles the old osccal value, which is in the last spot of programming memory in these super old chips.