data Stack a = Stack { run :: [a] -> (a, [a]) }

[...skipped...]

But, I have simply no clue how to fix that. :-(
Can anybody give my a hint?

Yes. It's simply impossible. The Stack data type can't be turned into a monad.

May be you can explain what do you want to do with this "monad"? What kind of code would you write if it would be such monad?
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