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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, $τ$, 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 $τ$ and the duration of reionization, $Δz$. We show that by combining the kSZ two-point function and the reconstructed kSZ four-point function, degeneracies between $τ$ and $Δz$ can be strongly broken, yielding tight constraints on both parameters. We forecast $σ(τ) = 0.003$ and $σ(Δz) = 0.25$ for a combination of CMB-S4 and Planck data, including detailed treatment of foregrounds and atmospheric noise. The constraint on $τ$ 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.