Provided your rope is in good shape, and not swollen or knotted somehow, either the rope pulley or recoil spring are broken or damaged and not allowing the recoil pressure to retract the rope. Unfortunately, there are no step-by-step instructions I am aware of for the repair as it typically would be done by a professional who would rely on their experience and training to determine the best method.
Due to the small size required for this application, there is no separate recoil bolted to the engine that can be easily replaced as one assembly. That means some of the engine assembly will need to come apart to get at and install the parts. In addition, the recoil is under a lot of spring pressure and the spring itself is a small band of metal thin enough to cut you, so if you have any reservation you might consider consulting a professional for the repair. If you do decide to tackle it yourself, be sure to take every possible safety precaution.
Checking the parts illustrations on the Sears PartsDirect site, or in the image below, will help give you some idea how the components fit and work together. The assembly you need to reach is shown as reference key 22. Remember it will be attached to key 23, the spring, and will be preloaded with pressure so it will be difficult to get apart. And donâ€™t forget to wind some pressure back in after installing the new parts so it will retract all the way when starting. Sometimes a deep socket will fit and can help spin the assembly to release or add spring tension.