Gridded sea surface height (SSH) maps estimated from satellite altimetry are widely used for estimating surface ocean geostrophic currents. Satellite altimeters observe SSH along one-dimensional tracks widely spaced in space and time, making accurately reconstructing the two-dimensional (2D) SSH field challenging. Traditionally, SSH is mapped using optimal interpolation (OI). However, OI artificially smooths the SSH field leading to high mapping errors in regions with rapidly-evolving mesoscale features such as western boundary currents. Motivated by the dynamical relation between SSH and sea surface temperature (SST) and the notion that even the chaotic evolution of mesoscale ocean turbulence may contain repeating patterns, we outline a deep learning (DL) approach where a neural network is trained to reconstruct 2D SSH by synthesizing altimetry and SST observations. In the Gulf Stream Extension region, dominated by mesoscale variability, our DL method substantially improves the SSH reconstruction compared to existing methods. Our SSH map has 17% lower root-mean-square error and resolves spatial scales 30% smaller than OI compared against independent altimeter observations. Surface geostrophic currents calculated from our map are closer to surface drifter observations and appear qualitatively more realistic, with stronger currents, a clearer separation between the Gulf Stream and neighboring eddies, and the appearance of smaller coherent eddies missed by other methods. Our map yields significant re-estimations of important dynamical quantities such as eddy kinetic energy, vorticity, and strain rate. Applying our DL method to produce a global SSH product may provide a more accurate and higher resolution product for studying mesoscale ocean turbulence.