= Solution
For the fixed degree-two <irreducible polynomial> $f$, a <matrix> with <minimal polynomial> a power of $f$ has only one primary component. Its <conjugacy classes> correspond to <integer partitions> $\lambda$ of $m$. By the <primary matrix centralizer formula>, its <centralizer> has exactly the same order $c_\lambda(q^2)$ as the <centralizer> of the corresponding <unipotent matrix> in $GL_m(q^2)$.
Sum the <conjugacy class> sizes:
$$
\#\{g\in GL_{2m}(q):\mu_g=f^a\text{ for some }a\ge1\}
=|GL_{2m}(q)|\sum_{\lambda\vdash m}\frac1{c_\lambda(q^2)}.
$$
The same sum times $|GL_m(q^2)|$ is the number of <unipotent matrices> of $GL_m(q^2)$, which 5(c) shows is $(q^2)^{m^2-m}$. Hence <counting matrices with a fixed primary polynomial> gives
$$
\boxed{q^{2m^2-2m}\frac{|GL_{2m}(q)|}{|GL_m(q^2)|}.}
$$
In particular the exponent $a$ of the minimal polynomial is not fixed in advance: summing all <integer partitions> includes every possible largest block size. The usual statement takes $m\ge1$; dimension zero can instead be included with its identity convention.
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