Supernovae that are strongly gravitationally lensed (gLSNe) by elliptical galaxies are powerful probes of astrophysics and cosmology that will be discovered systematically by wide-field, high-cadence imaging surveys such as the Zwicky Transient Facility (ZTF) and the Large Synoptic Survey Telescope (LSST). Here we use pixel-level simulations that include observing strategy, target selection, supernova properties, and dust to forecast the rates and properties of gLSNe that ZTF and LSST will find. Applying the resolution-insensitive discovery strategy of Goldstein et al., we forecast that ZTF (LSST) can discover 0.02 (0.79) 91bg-like, 0.17 (5.92) 91T-like, 1.22 (47.84) Type Ia, 2.76 (88.51) Type IIP, 0.31 (12.78) Type IIL, and 0.36 (15.43) Type Ib/c gLSNe per year, with uncertainties dominated by uncertainties in the supernova rate. We also forecast that the surveys can discover at least 3.75 (209.32) Type IIn gLSNe per year, for a total of at least 8.60 (380.60) gLSNe per year under fiducial observing strategies. ZTF gLSNe have a median z s = 0.9, z l = 0.35, , Δt max = 10 days, min(θ) = 0.″25, and N img = 4. LSST gLSNe are less compact and less magnified, with a median z s = 1.0, z l = 0.4, , Δt max = 25 days, min(θ) = 0.″6, and N img = 2. We develop a model of the supernova-host galaxy connection and find that the vast majority of gLSN host galaxies will be multiply imaged, enabling detailed constraints on lens models with sufficiently deep high-resolution imaging taken after the supernova has faded. We release the results of our simulations as catalogs at http://portal.nersc.gov/project/astro250/glsne/.