No problem! Usually the tip of the tool remains in the material being spot drilled. These pieces can be a real pain to get out. Center punch, carbide endmill or drilling a hole from the opposite side of the part. Instead, try grinding a slot right down the middle of the broken tool. This can be set up on a surface grinder with an abrasive cutoff wheel or if you have a steady hand, you can do it with the broken tool held in a vise and with a hand held grinder/abrasive cutoff wheel split the tool right down the middle. Look at the image below.

                           

When you have the tool prepped, place it back in the chuck or collet. Use the tool to drill around the broken tip. Usually you don't have to go very deep before you can break off the material along with the broken tip of the tool. This works on mills as well as lathes.

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