Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of Reionization II. The Deepest Near-Infrared Spectroscopic Observation at $z\gtrsim7

Realizing the utility of Ly$\alpha$ emission to trace the evolution of the intergalactic medium (IGM) during the epoch of reionization requires deep spectroscopy across the boundary of optical and near-infrared (NIR) spectrographs at $z\sim7.2$ when Ly$\alpha$ emission is at $\sim$1$\mu$m. Our Texas Spectroscopic Search for Ly$\alpha$ Emission at the End of Reionization includes 18 nights of deep spectroscopic observations using the Keck DEIMOS (optical) and MOSFIRE (NIR) spectrographs. Within this dataset we observe Ly$\alpha$ emission from 183 photometric-redshift selected galaxies at $z =$ 5.5 - 8.3 from the Cosmic Assembly Near infrared Deep Extragalactic Legacy Survey (CANDELS). Our overlapping MOSFIRE observations, over 84 galaxies total, provide the deepest NIR spectroscopic data yet obtained for Ly$\alpha$ from galaxies $z > 7$, with $>16$ hr integration time for four observed galaxies. Here we analyze these four targets, and we report the discovery of a new $z = 7.60$ Ly$\alpha$ detection as well as provide an updated observation of the previously confirmed $z=7.51$ Ly$\alpha$ emission from Finkelstein et al. (2013) with a $\sim$3$\times$ longer exposure time. Our analysis of these Ly$\alpha$ emission line profiles reveal a significant asymmetric shape. The two detected Ly$\alpha$ emission lines from bright sources ($M_{\text{UV}}