The pilot said that on short final to the airstrip he encountered a windshear and downdraft. He added power to go around and the plastic control wheel on the left side broke into two sections at the middle. The aircraft hit the ground hard and veered off the runway. The pilot stated that after his control wheel broke, he attempted to control the aircraft by grabbing the wheel on the right side of the cockpit; however, the aircraft hit the ground and he lost directional control. The plastic control wheels in this aircraft are the subject of a reoccurring airworthiness directive (69-22-2), which requires repetitive dye penetrant inspections at each annual inspection for cracks in the center of the wheel. The maintenance records disclosed that the control wheel was inspected in accordance with the AD at the last annual. Microscopic examination of the fracture face revealed no evidence of dye penetrant, indicating that the crack did not exist at the last inspection.