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 similar to 7-10 h, we detect no Ly alpha emission with signal-to-noise ratio (S/N) > 5 in our sample. We determine our observations to be 80 per cent complete for 5 sigma spatially and spectrally unresolved emission lines with integrated line flux > 5.7 x 10(-18) erg s(-1) cm(-2). We define a photometrically selected sub-sample of 29 targets at z = 7.9 +/- 0.6, with a median 5 sigma Ly alpha EW limit of 58 angstrom. 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 per cent), > 0.46 (95 per cent) at z similar to 8, providing further evidence of rapid reionization at z similar to 6-8. We show that this is consistent with reionization history models extending the galaxy luminosity function to M-UV less than or similar to -12, with low ionizing photon escape fractions, f(esc) less than or similar to 15 per cent.