  JWST has recently sparked a new era of Ly$\alpha$ spectroscopy, delivering
the first measurements of the Ly$\alpha$ escape fraction and velocity profile
in typical galaxies at $z\simeq6-10$. These observations offer new prospects
for insight into the earliest stages of reionization. But to realize this
potential, we need robust models of Ly$\alpha$ properties in galaxies at
$z\simeq5-6$ when the IGM is mostly ionized. Here we use new JWST observations
from the JADES and FRESCO surveys combined with VLT/MUSE and Keck/DEIMOS data
to characterize statistical distributions of Ly$\alpha$ velocity offsets,
escape fractions, and EWs in $z\simeq5-6$ galaxies. We find that galaxies with
large Ly$\alpha$ escape fractions (> 0.2) are common at $z\simeq5-6$,
comprising 30 per cent of Lyman break selected samples. Comparing to literature
studies, our census suggests that Ly$\alpha$ becomes more prevalent in the
galaxy population toward higher redshift from $z\sim3$ to $z\sim6$, although we
find that this evolution slows considerably between $z\sim5$ and $z\sim6$,
consistent with modest attenuation from residual HI in the mostly ionized IGM
at $z\simeq5-6$. We find significant evolution in Ly$\alpha$ velocity profiles
between $z\simeq2-3$ and $z\simeq5-6$, likely reflecting the influence of
resonant scattering from residual intergalactic HI on the escape of Ly$\alpha$
emission near line center. This effect will make it challenging to use
Ly$\alpha$ peak offsets as a probe of Lyman continuum leakage at $z\simeq5-6$.
We use our $z\simeq5-6$ Ly$\alpha$ distributions to make predictions for
typical Ly$\alpha$ properties at $z\gtrsim8$ and discuss implications of a
recently-discovered Ly$\alpha$ emitter at $z\simeq8.5$ with a small peak
velocity offset (156 km s$^{-1}$).
