New answers tagged convergence
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It turns out the issue was very simple (and not related to a limitation of numpy's matrix power function as such). I initially thought there was some numerical floating point error being propagated - but the example matrix I was testing on contains only integers. The problem was that at $k=20$ the value of the matrix entries exceeded numpy's maximum possible ...
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The Frobenius norm is not an operator norm, it is a norm on the vector space of linear operators/matrices, which is not the same thing. Just change it to any other preset norm and it should work.
It is also the case that your method of computing matrix powers is not stable. The algorithm used in Numpy is basic repeated squaring, which has no normalization ...
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