I can't find how to hard reset the mini with the hardware
anywhere.
I assume you've tried plugging the mini in? Sometimes that gives it a boost and makes it start up.
And, have you tried plugging it in and
then doing all the reset options (so, soft reset by putting a pin in the whole, and harder reset by sliding & holding until off, then putting and holing pin in hole, then sliding power on)?
If nothing works I'd ask Kobo directly for the hard reset with hardware. And if they say there isn't one you'll have to ask a kind person with a mini to give you an image of "the" partition... There doesn't seem to be one circulating yet for the mini.
Edit: Oh, there are people that actually have an image:
https://www.mobileread.com/forums/sho...d.php?t=193321
https://www.mobileread.com/forums/sho...d.php?t=193411