The rapidly increasing number of Internet of things devices poses significant challenges to the service capabilities of terrestrial networks. Ultra-dense low earth orbit satellite constellations have emerged as a promising solution to address the limited coverage area of terrestrial networks and the insufficient computing capabilities of user terminals. By establishing a satellite-to-terrestrial connection model, the average computational resources and backhaul capacity available to ground users were analyzed and a multi-objective optimization problem for designing ultra-dense LEO constellations was modelled, aiming to minimize the total number of satellites while meeting the heterogeneous quality of service (QoS) requirements for user task offloading. A two-stage constellation design algorithm was proposed, which firstly optimized the resource allocation strategy of computational satellites and then determined the constellation parameters under optimal QoS conditions. Simulation results validate the accuracy of the theoretical model and analyze the impact of orbital altitude, task volume, and coverage requirements on constellation design. Compared with the Kuiper, OneWeb, and SpaceX constellations, the proposed constellation achieved an average coverage rate increase of 29.52%, 48.73%, and 34.82%, respectively, with the same number of satellites.