Ice-penetrating radar data contain a wealth of information about the bed and internal structure of the ice sheet. While these data have long been used to diagnose the presence of basal water, infer attenuation rates, or explore the internal stratigraphy of the ice sheet, they have rarely been used jointly in a formal inverse model for the ice sheet temperature structure. Here, we invert a coupled thermomechanical ice sheet and basal hydrology model to infer both geothermal heat flow (GHF) and accumulation rate from multiple classes of radar observations in the area around Dome A, East Antarctica. Our forward model solves for a coupled steady state between the ice sheet flow field, temperature, and basal hydrology, including melt, water transport, and freeze-on. We fit radar observations of basal water, freeze-on, and internal layers, along with a GHF prior based on aeromagnetic observations. We minimize the combined misfit function by first using an evolutionary algorithm followed by localized perturbation tests. In addition to inferring the spatial distribution of GHF and accumulation rate, we are also able to estimate the uncertainty about our best-fit answer, as well as quantify how our result depends on each individual data constraint. Our results demonstrate a new method for combining multiple glaciological constraints into a single inverse model of the ice sheet, and give us a more rigorous picture of the information content provided by each data set. In a companion paper we analyze and interpret the best-fit model.