The ultraviolet (UV) spectral energy distributions of low-mass (K- and M-type) stars play a critical role in the heating and chemistry of exoplanet atmospheres, but are not observationally well-constrained. Direct observations of the intrinsic flux of the Lyman alpha line (the dominant source of UV photons from low-mass stars) are challenging, as interstellar HI absorbs the entire line core for even the closest stars. To address the existing gap in empirical constraints on the UV flux of K and M dwarfs, the MUSCLES HST Treasury Survey has obtained UV observations of 11 nearby M and K dwarfs hosting exoplanets. This paper presents the Lyman alpha and extreme-UV spectral reconstructions for the MUSCLES targets. Most targets are optically inactive, but all exhibit significant UV activity. We use a Markov Chain Monte Carlo technique to correct the observed Lyman alpha profiles for interstellar absorption, and we employ empirical relations to compute the extreme-UV spectral energy distribution from the intrinsic Lyman alpha flux in ~100 {AA} bins from 100-1170 {AA}. The reconstructed Lyman alpha profiles have 300 km/s broad cores, while >1% of the total intrinsic Lyman alpha flux is measured in extended wings between 300 km/s to 1200 km/s. The Lyman alpha surface flux positively correlates with the MgII surface flux and negatively correlates with the stellar rotation period. Stars with larger Lyman alpha surface flux also tend to have larger surface flux in ions formed at higher temperatures, but these correlations remain statistically insignificant in our sample of 11 stars. We also present HI column density measurements for 10 new sightlines through the local interstellar medium.
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