Place a keyhole hacksaw across one side of the nut as close to the thread as possible.
Hold the hacksaw as close to a 90-degree angle as you can, and ensure the saw's teeth don't come into contact with the thread. The goal is to cut off the broken nut without damaging the thread.
Saw back and forth across the nut using small movements. Keep the saw in the same groove, and stop sawing just before you have sawed completely through the nut. Use a pair of pliers to twist the nut off the threads so that you don't damage the surface under the nut with the saw.