Is there a tool that will remove a bolt when the head has been broken off
Yes. There are several ways to do it. If the bold is sticking out you could try getting a pair of alligator pliers. I think true value carries them. They work great on broken bolts they tighten more and more the harder you turn them. Never use Vise Grips you will only hurt yourself. If the bolt is broken off flush with the surface is is a more difficult task. You can try an "easy-out" also called a screw extractor. They are used by drilling a small hole through the center of the bolt. Then you drive the extractor into the hole. Then you use a wrench or pliers to turn out the bolt. Some helpful tips include using liquid wrench or some other penetrant before you start. Really stubborn bolts sometimes need some heat to free them up. A bernzomatic torch works well but make sure you are in a safe place to works and use fire safe practices and precautions while doing this.