  We consider the two-loop corrections to the $HW^+W^-$ vertex at order
$\alpha\alpha_s$. We construct a canonical basis for the two-loop integrals
using the Baikov representation and the intersection theory. By solving the
$\epsilon$-form differential equations, we obtain fully analytic expressions
for the master integrals in terms of multiple polylogarithms, which allow fast
and accurate numeric evaluation for arbitrary configurations of external
momenta. We apply our analytic results to the decay process $H \to \nu_e e W$,
and study both the integrated and differential decay rates. Our results can
also be applied to the Higgs production process via $W$ boson fusion.
