Well, it is pretty clear that $\mathbb{Q}(\pi^2-2\pi+5)$ are all algebraic. I don't have any idea how to justify it, but I think those are the only algebraic elements of $\mathbb{Q}(\pi)$ over $\mathbb{Q}(\pi^2-2\pi+5)$, yet I don't succeed to show it or recall a theorem which might help.
I would also like an approach for the general case, when relating $\mathbb{Q}(\pi)$ as $\mathbb{Q}(x)$ and asking the same question regarding a general polynomial $g(x)$.
$\pi$ is algebraic, as it is the root of $X^2-2X+5 - (\pi^2-2\pi + 5)$.
Then all polynomials in $\pi$ are algebraic, because algebraic numbers are stable under product and sum. Moreover, nonzero algebraic numbers are stable under inverses, so all rational fractions in $\pi$ are algebraic : $\mathbb{Q}(\pi)/\mathbb{Q}(\pi^2-2\pi+5)$ is an algebraic extension.