My iPad mini 2 won't update.
No, the iPad Mini 2 is eligible for an upgrade to iOS 10.
Have you tried upgrading via iTunes? It could be that your device doesn't have enough storage and is failing to update because of it.
If that doesn't work, you can make a back-up (either by connecting your iPad to iTunes, or backing up to iCloud), and then restore the iPad. Just connect your iPad to iTunes, go to 'summary' and then click 'restore'.