  Patchy reionization leaves a number of imprints on the small-scale cosmic
microwave background (CMB) temperature fluctuations, the largest of which is
the kinematic Sunyaev-Zel'dovich (kSZ), the Doppler shift of CMB photons
scattering off moving electrons in ionized bubbles. It has long been known that
in the CMB power spectrum, this imprint of reionization is largely degenerate
with the kSZ signal produced by late-time galaxies and clusters, thus limiting
our ability to constrain reionization. Following Smith & Ferraro (2017), it is
possible to isolate the reionization contribution in a model independent way,
by looking at the large scale modulation of the small scale CMB power spectrum.
In this paper we extend the formalism to use the full shape information of the
small scale power spectrum (rather than just its broadband average), and argue
that this is necessary to break the degeneracy between the optical depth $\tau$
and parameters setting the duration of reionization. In particular, we show
that the next generation of CMB experiments could achieve up to a factor of 3
improvement on the optical depth $\tau$ and at the same time, constrain the
duration of reionization to $\sim$ 25 %. This can help tighten the constrains
on neutrino masses, which will be limited by our knowledge of $\tau$, and shed
light on the physical processes responsible for reionization.
