There's an easy way to work out the distance to the mirror at the present time, if it moves with the Hubble flow: imagine that instead of being emitted by us, the light is emitted by the mirror image of the Earth and simply passes through the mirror to us. The farthest comoving distance that the mirror Earth could be is the same as the farthest present-day distance from which light emitted in the present day can ever reach us; assuming ΛCDM is correct, that's around 16.5 billion light years. The comoving distance to the mirror (and the metric distance to it in the present day) is exactly half that.
Working out the metric distance to the mirror at the time of reflection is trickier. If (nonstandard terminology warning) $D(a)$ is the comoving distance light can travel from scale factor $a$ to $∞$, then the scale factor at which the light hits the mirror is $D^{-1}\left(\frac12 D(1)\right)$. If all energy was dark energy and the expansion was exactly exponential then the solution would be $2$ exactly. Using central parameters from here I get $a\approx 2.09$, so the mirror distance is a bit higher, around 17 billion light years.
For what it's worth there's an exact expression for this function in a flat perfect dust+dark energy universe:
$$D(a) = \frac{c}{a \sqrt{Ω_Λ} H_0} \; {}_2F_1\left(\frac13,\frac12;\frac43;-\frac{Ω_m}{a^3Ω_Λ}\right)$$
where ${}_2F_1$ is the hypergeometric function.
