We present the results of an imaging observation campaign conducted with the Subaru Telescope adaptive optics system (IRCS+AO188) on 28 gravitationally lensed quasars and candidates (23 doubles, 1 quad, 1 possible triple, and 3 candidates) from the SDSS Quasar Lens Search. We develop a novel modelling technique that fits analytical and hybrid point spread functions (PSFs), while simultaneously measuring the relative astrometry, photometry, as well as the lens galaxy morphology. We account for systematics by simulating the observed systems using separately observed PSF stars. The measured relative astrometry is comparable with that typically achieved with the Hubble Space Telescope, even after marginalizing over the PSF uncertainty. We model for the first time the quasar host galaxies in five systems, without a priori knowledge of the PSF, and show that their luminosities follow the known correlation with the mass of the supermassive black hole. For each system, we obtain mass models far more accurate than those previously published from low-resolution data, and we show that in our sample of lensing galaxies the observed light profile is more elliptical than the mass, for ellipticity >~0.25. We also identify eight doubles for which the sources of external and internal shear are more reliably separated, and should therefore be prioritized in monitoring campaigns aimed at measuring time delays in order to infer the Hubble constant.