I cannot update my Mac Mini (Mid 2010) past High Sierra 10. 13. 6. What hardware do I need to change?

sorry, but once a Mac is no longer eligible for macOS upgrades, there are NO available hardware upgrades that will make it once again eligible for further macOS upgrades.

You basically need a brand new mac.

if it works well, just keep it on your network, use it for a child for school, store stuff (but ensure you make copy of important stuff before it crawls completely).

No worries to keep it as long as it works, the beautiful thing with the mini is… it’s mini and does not requires lots of real estate to keep it. I had one like this which was almost good for garbage about 6-7 years ago accorfing to Apple Store, and with a tiny welding in an Apple approved shop, made those 7 more years, among which 2 of sudden school at home with pandemic.

It’s not because it does not provide latest macOS capabilities that it becomes useless.