The Sterile Processing and Distribution (SPD) department is responsible for cleaning, disinfecting, inspecting, and assembling surgical instruments between surgeries. Manual inspection and preparation of instrument trays is a time-consuming, error-prone task, often prone to contamination and instrument breakage. In this work, we present a fully automated robotic system that sorts and structurally packs surgical instruments into sterile trays, focusing on automation of the SPD assembly stage. A custom dataset comprising 31 surgical instruments and 6,975 annotated images was collected to train a hybrid perception pipeline using YOLO12 for detection and a cascaded ResNet-based model for fine-grained classification. The system integrates a calibrated vision module, a 6-DOF Staubli TX2-60L robotic arm with a custom dual electromagnetic gripper, and a rule-based packing algorithm that reduces instrument collisions during transport. The packing framework uses 3D printed dividers and holders to physically isolate instruments, reducing collision and friction during transport. Experimental evaluations show high perception accuracy and statistically significant reduction in tool-to-tool collisions compared to human-assembled trays. This work serves as the scalable first step toward automating SPD workflows, improving safety, and consistency of surgical preparation while reducing SPD processing times.