The epoch of reionization is one of the major phase transitions in the
history of the universe, and is a focus of ongoing and upcoming cosmic
microwave background (CMB) experiments with improved sensitivity to small-scale
fluctuations. Reionization also represents a significant contaminant to
CMB-derived cosmological parameter constraints, due to the degeneracy between
the Thomson-scattering optical depth, $\tau$, and the amplitude of scalar
perturbations, $A_s$. This degeneracy subsequently hinders the ability of
large-scale structure data to constrain the sum of the neutrino masses, a major
target for cosmology in the 2020s. In this work, we explore the kinematic
Sunyaev-Zel'dovich (kSZ) effect as a probe of reionization, and show that it
can be used to mitigate the optical depth degeneracy with high-sensitivity,
high-resolution data from the upcoming CMB-S4 experiment. We discuss the
dependence of the kSZ power spectrum on physical reionization model parameters,
as well as on empirical reionization parameters, namely $\tau$ and the duration
of reionization, $\Delta z$. We show that by combining the kSZ two-point
function and the reconstructed kSZ four-point function, degeneracies between
$\tau$ and $\Delta z$ can be strongly broken, yielding tight constraints on
both parameters. We forecast $\sigma(\tau) = 0.003$ and $\sigma(\Delta z) =
0.25$ for a combination of CMB-S4 and Planck data, including detailed treatment
of foregrounds and atmospheric noise. The constraint on $\tau$ is nearly
identical to the cosmic-variance limit that can be achieved from large-angle
CMB polarization data. The kSZ effect thus promises to yield not only detailed
information about the reionization epoch, but also to enable high-precision
cosmological constraints on the neutrino mass.