= Solution
Write $n_b=n_p+n_H$ for the <cosmological baryon density> in this hydrogen-only model, and $x_e=n_p/n_b$ for the <hydrogen ionization fraction>. By <charge neutrality>, $n_e=n_p$, so the <Saha ionization equation> becomes
$$
\frac{1-x_e}{x_e^2}=n_b\left(\frac{2\pi}{M_eT}\right)^{3/2}e^{B/T}.
$$
The <baryon-to-photon ratio> is $\eta=n_b/n_\gamma$, and the equilibrium <photon number density> is $n_\gamma=2\zeta(3)T^3/\pi^2$. At an order-one <hydrogen ionization fraction>, neglecting factors of order one therefore gives
$$
1\simeq\eta\left(\frac T{M_e}\right)^{3/2}e^{B/T},\qquad T\simeq\frac{B}{\log[\eta^{-1}(M_e/T)^{3/2}]}.
$$
Replacing $T$ inside the slowly varying logarithm by $B$ gives
$$
\boxed{T\simeq\frac{B}{\log[\eta^{-1}(M_e/B)^{3/2}]}\ll B.}
$$
With $B=13.6\,\mathrm{eV}$, $M_e=5\times10^5\,\mathrm{eV}$ and $\eta=10^{-10}$, this is $T\simeq0.351\,\mathrm{eV}\simeq4.1\times10^3\,\mathrm K$. Solving the preceding implicit approximation instead gives $T\simeq0.306\,\mathrm{eV}\simeq3.5\times10^3\,\mathrm K$. Both are reasonable estimates of the <recombination temperature> at the accuracy requested. The <small baryon abundance delays hydrogen recombination>: even when $T\ll B$, the high-energy tail of the abundant <photons> can still ionize the much rarer <hydrogen atoms>.
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