Use a rotary tool with a grinding tip or cutting wheel to get the head of the broken bolt as close to flat as possible. Don't smooth it down to the point where your bit cannot grab the material, but you do want a flat spot that will allow you to drill a pilot hole. Be careful not to damage the tile or shower surround while you work. Once the head is flattened, use a punch or nail to make a small indentation in the center of the bolt head. This will make drilling your pilot hole much easier.
Drill a pilot hole in the bolt with a small left-handed bit and a drill that is reversible. Using a left-handed bit will avoid further tightening down of the bolt by drilling in a clockwise direction. Drill in reverse. In some lucky circumstances, the bolt will come loose from the torque. Be careful, as the drill bit will have a tendency to slip off the bolt head and easily can chip a ceramic tile or fiberglass surround.
After your pilot hole is drilled, spray chemical lubricant or silicone spray on the bolt and allow it to sit overnight. Ideally, you want the bolt to come loose before you have to physically drill it out of the threaded hole entirely. A little lubricant can help loosen rust and stuck threads, and might just save you a lot of work.
After your pilot hole is drilled, use a larger left-handed bit to widen the hole. Make sure the bit still is smaller than the bolt itself to avoid damaging the threads of the hole. If the bolt does not come loose during the first drilling, reverse your drill, remove the bit from the bolt hole and drill it again. Repeat this process a few times and the bolt should come loose.