Murphy's Law guarantees that screws will break at the worst times. Fortunately, I've found a simple way to extract the headless shank and repair the damage. I use the countersink collar from one of my drill-and-countersink sets. Make sure the countersink's bore is slightly larger than the buried shank. Use the countersink to drill around the broken screw. If drilling alone doesn't free the shank, the oversize hole creates room for needle-nose pliers to grip and turn. After the broken screw has been removed, glue in a plug cut from the face of another board. Face-grain plugs hold screws much better than end-grain dowels. Pare the plug flush with the mortise, and you're back in business.