Lyα is the transition to the ground state from the first excited state of hydrogen (the most common element). Resonant scattering of this line by neutral hydrogen greatly impedes its emergence from galaxies, so the fraction of galaxies emitting Lyα is a tracer of the neutral fraction of the intergalactic medium (IGM), and thus the history of reionisation. In previous works, we used early JWST/NIRSpec data from the JWST Advanced Deep Extragalactic Survey (JADES) to classify and characterise Lyα emitting galaxies (LAEs). This survey is approaching completion, and the current sample is nearly an order of magnitude larger. From a sample of 795 galaxies in JADES at 4.0 < z < 14.3, we find evidence for Lyα emission in 150 sources. We reproduce the previously found correlation between Lyα escape fraction ($f\rm _{esc}^{Ly\alpha }$) - Lyα rest-frame equivalent width (REWLyα) and the negative correlation between Lyα velocity offset - $f\rm _{esc}^{Ly\alpha }$. Both $f\rm _{esc}^{Ly\alpha }$ and REWLyα decrease with redshift (z ≳ 5.5), indicating the progression of reionisation on a population scale. Our data are used to demonstrate an increasing IGM transmission of Lyα from z ∼ 14 − 6. We measure the completeness-corrected fraction of LAEs ($X\rm _{Ly\alpha }$) from z = 4 − 9.5. An application of these $X\rm _{Ly\alpha }$ values to the results of previously utilised semi-analytical models suggests a high neutral fraction at z = 7 (XHI ∼ 0.8 − 0.9). Using an updated fit to the intrinsic distribution of REWLyα results in a lower value in agreement with current works (${X_{\rm HI}}= 0.64_{-0.21}^{+0.13}$). This sample of LAEs will be paramount for unbiased population studies of galaxies in the EoR.