  Detections and non-detections of Lyman alpha (Ly$\alpha$) emission from $z>6$
galaxies ($<1$ Gyr after the Big Bang) can be used to measure the timeline of
cosmic reionization. Of key interest to measuring reionization's mid-stages,
but also increasing observational challenge, are observations at z > 7, where
Ly$\alpha$ redshifts to near infra-red wavelengths. Here we present a search
for z > 7.2 Ly$\alpha$ emission in 53 intrinsically faint Lyman Break Galaxy
candidates, gravitationally lensed by massive galaxy clusters, in the KMOS
Lens-Amplified Spectroscopic Survey (KLASS). With integration times of ~7-10
hours, we detect no Ly$\alpha$ emission with S/N>5 in our sample. We determine
our observations to be 80% complete for 5$\sigma$ spatially and spectrally
unresolved emission lines with integrated line flux $>5.7\times10^{-18}$ erg
s$^{-1}$ cm$^{-2}$. We define a photometrically selected sub-sample of 29
targets at $z=7.9\pm0.6$, with a median 5$\sigma$ Ly$\alpha$ EW limit of 58A.
We perform a Bayesian inference of the average intergalactic medium (IGM)
neutral hydrogen fraction using their spectra. Our inference accounts for the
wavelength sensitivity and incomplete redshift coverage of our observations,
and the photometric redshift probability distribution of each target. These
observations, combined with samples from the literature, enable us to place a
lower limit on the average IGM neutral hydrogen fraction of $> 0.76 \; (68\%),
\; > 0.46 \; (95\%)$ at z ~ 8, providing further evidence of rapid reionization
at z~6-8. We show that this is consistent with reionization history models
extending the galaxy luminosity function to $M_\textrm{UV} \lesssim -12$, with
low ionizing photon escape fractions, $f_\textrm{esc} \lesssim 15\%$.
