Inter-satellite cooperative computing empowers satellite edge computing to transcend the limitations of constrained onboard computing power, ushering in a new paradigm for in-orbit computation. However, within the large-scale LEO space environment characterized by abrupt failures, dynamic topologies and constrained resources, inter-satellite cooperative computing faces threats of reduced reliability and inefficiency. In this paper, we delve into the joint optimization of resilient computing networking, computation offloading, and resource allocation, with the goal of maximizing the quality of service for computing requests. Specifically, we propose a novel resilient inter-satellite computing collaboration architecture that enhances the reliability and serviceability of inter-satellite computing in unstable network space environments. Furthermore, we develop a trust-region-based multi-agent deep reinforcement learning algorithm to efficiently enable autonomous collaborative computation among satellites in a large-scale LEO space environment in a distributed manner. Finally, the effectiveness and superiority of our approach are validated through extensive simulation experiments.