  We report results from an analysis aimed at detecting the trispectrum of the
kinematic Sunyaev-Zel{'}dovich (kSZ) effect by combining data from the South
Pole Telescope (SPT) and {\it Herschel}-SPIRE experiments over a 100 ${\rm
deg}^{2}$ field. The SPT observations combine data from the previous and
current surveys, namely SPTpol and SPT-3G, to achieve depths of 4.5, 3, and 16
$\mu {\rm K-arcmin}$ in bands centered at 95, 150, and 220 GHz. For SPIRE, we
include data from the 600 and 857 GHz bands. We reconstruct the
velocity-induced large-scale correlation of the small-scale kSZ signal with a
quadratic estimator that uses two cosmic microwave background (CMB) temperature
maps, constructed by optimally combining data from all the frequency bands. We
reject the null hypothesis of a zero trispectrum at $10.3\sigma$ level.
However, the measured trispectrum contains contributions from both the kSZ and
other undesired components, such as CMB lensing and astrophysical foregrounds,
with kSZ being sub-dominant. We use the \textsc{Agora} simulations to estimate
the expected signal from CMB lensing and astrophysical foregrounds. After
accounting for the contributions from CMB lensing and foreground signals, we do
not detect an excess kSZ-only trispectrum and use this non-detection to set
constraints on reionization. By applying a prior based on observations of the
Gunn-Peterson trough, we obtain an upper limit on the duration of reionization
of $\Delta z_{\rm re, 50} < 4.5$ (95\% C.L). We find these constraints are
fairly robust to foregrounds assumptions. This trispectrum measurement is
independent of, but consistent with, {\it Planck}'s optical depth measurement.
This result is the first constraint on the epoch of reionization using the
non-Gaussian nature of the kSZ signal.
