= Solution
In a large volume, one one-particle state occupies momentum-space volume $h^3/V$. A spherical shell of radius $p$ has volume $4\pi p^2dp$, so for spin degeneracy $g$ the <density of states> is $4\pi gVp^2dp/h^3$. With ultra-relativistic dispersion $E=cp$ and zero <chemical potential>, the <Bose-Einstein distribution> gives
$$
\epsilon=\frac{4\pi gc}{h^3}\int_0^\infty\frac{p^3\,dp}{e^{cp/(kT)}-1}
=\frac{4\pi g(kT)^4}{h^3c^3}\int_0^\infty\frac{x^3\,dx}{e^x-1}.
$$
The convergent dimensionless integral is $\pi^4/15$, obtained by expansion into exponentials and summing $\sum n^{-4}$. Therefore \b[$\epsilon\propto T^4$ and $\alpha=4$]. Photons are massless bosons with two polarizations; equilibrium emission and absorption do not conserve photon number and force $\mu_\gamma=0$.
For recombination, <chemical equilibrium> imposes $\mu_p+\mu_e=\mu_H+\mu_\gamma=\mu_H$, including rest <energies> in the <chemical potentials>. Divide the three nonrelativistic number-density expressions to obtain
$$
\frac{n_pn_e}{n_H}
=\frac{g_pg_e}{g_H}\left(\frac{2\pi kT}{h^2}\right)^{3/2}
\left(\frac{m_pm_e}{m_H}\right)^{3/2}e^{-I/(kT)}.
$$
Here $m_pc^2+m_ec^2-m_Hc^2=I$. Neglect the relative proton/hydrogen mass difference and take the ground-state degeneracy ratio $g_pg_e/g_H=1$ (<Electron> and proton spins give $2\cdot2$ and hydrogen gives $4$). <Charge neutrality> gives $n_p=n_e$. This yields the stated <Saha equation>
$$
\boxed{\frac{n_e^2}{n_H}
=\left(\frac{2\pi m_ekT}{h^2}\right)^{3/2}e^{-I/(kT)}}.
$$
With $n_e=X_en_B$ and $n_H=(1-X_e)n_B$, the ratio asked for is
$$
\boxed{\frac{1-X_e}{X_e^2}
=\eta\,16\pi\zeta(3)\left(\frac{kT}{hc}\right)^3
\left(\frac{h^2}{2\pi m_ekT}\right)^{3/2}e^{I/(kT)}}.
$$
Equivalently the prefactor is $16\pi\zeta(3)\eta(2\pi)^{-3/2}[kT/(m_ec^2)]^{3/2}$. Because photons vastly outnumber baryons, even a small high-energy tail can keep hydrogen ionized. Neutralization therefore requires exponential suppression of that tail far below $kT=I$, explaining the much lower recombination temperature. The equilibrium formula is an estimate; the actual expanding-universe recombination kinetics eventually depart from Saha equilibrium.
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