incorrigibility - ist natürlich kein 100% Treffer

"unteachability" works ok... but even as a native AE speaker, I think this would probably be rendered differently in speech/writing.

For example: The unteachability of today's youth is no surprise.

That's perfectly ok, but it might be more common to see/read:
"The inability to teach today's youth is no surprise."

I see that LEO offers "unteachable" or "unconvincable"...
I don't think I have ever used (or would ever use) "unconvincability" in a sentence or in speech, but, I don't think that's a Quatschwort

hope that helps.