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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 ofcosmic reionization. Of key interest to measuring reionization's mid-stages,but also increasing observational challenge, are observations at z > 7, whereLy$\alpha$ redshifts to near infra-red wavelengths. Here we present a searchfor z > 7.2 Ly$\alpha$ emission in 53 intrinsically faint Lyman Break Galaxycandidates, gravitationally lensed by massive galaxy clusters, in the KMOSLens-Amplified Spectroscopic Survey (KLASS). With integration times of ~7-10hours, we detect no Ly$\alpha$ emission with S/N>5 in our sample. We determineour observations to be 80% complete for 5$\sigma$ spatially and spectrallyunresolved emission lines with integrated line flux $>5.7\times10^{-18}$ ergs$^{-1}$ cm$^{-2}$. We define a photometrically selected sub-sample of 29targets 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 thewavelength sensitivity and incomplete redshift coverage of our observations,and the photometric redshift probability distribution of each target. Theseobservations, combined with samples from the literature, enable us to place alower limit on the average IGM neutral hydrogen fraction of $> 0.76 \; (68\%),\; > 0.46 \; (95\%)$ at z ~ 8, providing further evidence of rapid reionizationat z~6-8. We show that this is consistent with reionization history modelsextending the galaxy luminosity function to $M_\textrm{UV} \lesssim -12$, withlow ionizing photon escape fractions, $f_\textrm{esc} \lesssim 15\%$.

Inferences on the Timeline of Reionization at z~8 From the KMOS Lens-Amplified Spectroscopic Survey