Converting process sketches into executable simulation models remains a major bottleneck in process systems engineering, requiring substantial manual effort and simulator-specific expertise. Recent advances in generative AI have improved both engineering-diagram interpretation and LLM-assisted flowsheet generation, but these remain largely disconnected: diagram-understanding methods often stop at extracted graphs, while text-to-simulation workflows assume structured inputs rather than raw visual artifacts. To bridge this gap, we present an end-to-end multi-agent large language model system that converts process diagrams directly into executable Aspen HYSYS flowsheets. The framework decomposes the task into three coordinated layers: diagram parsing and interpretation, simulation model synthesis, and multi-level validation. Specialized agents handle visual interpretation, graph-based intermediate representation construction, code generation for the HYSYS COM interface, execution, and structural verification. We evaluate the framework on four chemical engineering case studies of increasing complexity, from a simple desalting process to an industrial aromatic production flowsheet with multiple recycle loops. The system produces executable HYSYS models in all cases, achieving complete structural fidelity on the two simpler cases and strong performance on the more complex ones, with connection consistency above 0.93 and stream consistency above 0.96. These results demonstrate a viable end-to-end sketch-to-simulation workflow while highlighting remaining challenges in dense recycle structures, implicit diagram semantics, and simulator-interface constraints.