  The HI Ly$\alpha$ (1215.67 $\unicode{xC5}$) emission line dominates the
far-UV spectra of M dwarf stars, but strong absorption from neutral hydrogen in
the interstellar medium makes observing Ly$\alpha$ challenging even for the
closest stars. As part of the Far-Ultraviolet M-dwarf Evolution Survey (FUMES),
the Hubble Space Telescope has observed 10 early-to-mid M dwarfs with ages
ranging from $\sim$24 Myr to several Gyrs to evaluate how the incident UV
radiation evolves through the lifetime of exoplanetary systems. We reconstruct
the intrinsic Ly$\alpha$ profiles from STIS G140L and E140M spectra and achieve
reconstructed fluxes with 1-$\sigma$ uncertainties ranging from 5% to a factor
of two for the low resolution spectra (G140L) and 3-20% for the high resolution
spectra (E140M). We observe broad, 500-1000 km s$^{-1}$ wings of the Ly$\alpha$
line profile, and analyze how the line width depends on stellar properties. We
find that stellar effective temperature and surface gravity are the dominant
factors influencing the line width with little impact from the star's magnetic
activity level, and that the surface flux density of the Ly$\alpha$ wings may
be used to estimate the chromospheric electron density. The Ly$\alpha$
reconstructions on the G140L spectra are the first attempted on
$\lambda/\Delta\lambda\sim$1000 data. We find that the reconstruction precision
is not correlated with SNR of the observation, rather, it depends on the
intrinsic broadness of the stellar Ly$\alpha$ line. Young, low-gravity stars
have the broadest lines and therefore provide more information at low spectral
resolution to the fit to break degeneracies among model parameters.
