The correct way to fix a broken ice maker depends on how it is malfunctioning. If it is not making ice, check the supply line, water line and the solenoid. If it does not turn off, you may need to replace the ice maker.
The water supply may be the problem if ice maker's arm swings and makes noise for approximately 10 seconds while not producing ice. Fix this by removing any kinks in the water supply line and ensuring that it is not underneath the refrigerator.
The water line that comes into the freezer from behind may be blocked. Check for this problem by finding and turning off the water shut-off valve, unscrewing the copper line behind the refrigerator, and putting the line in a bucket. Then, turn on the valve, and examine if water comes out to verify if it was previously blocked.
The solenoid on the back of the refrigerator that the water line attaches to may be defective. If so, remove the sediment screen inside the solenoid, and flush it with water to clean it, which also offers an opportunity to ensure that no seals or diaphragms are damaged. Be sure to unplug the refrigerator before doing any of this.