Occasionally a drill bit (or the head of a bur bit) will break off inside the piece of precious metal. If the bit is protruding enough that you can grasp it with a pair if pliers, you can attempt to slowly twist it out. However, if it is lodged inside the piece and you can't get it out...never fear, ALUM is here!
Alum is a double sulfate chemical compound: a combination of an alkali metal, such as sodium, potassium, or ammonium and a trivalent metal, such as aluminum, iron, or chromium. It is commonly used in food processing (it makes pickles crispy) or as an astringent in medical applications. Alum is a non-acidic product that will not give off noxious fumes or burn holes in clothing, however if large amounts are ingested it can be fatal....so don't sprinkle it in your coffee and keep it away from children! It can be easily found in the canning or spice aisle of your grocery store.
Dissolve one tablespoon of alum in 1/4 cup of warm water in a glass jar. The alum solution attacks the ferrous metal, thus deteriorating the steel without harming the non-ferrous jewelry. Soak your piece in this solution and wait....it could take hours. The warmer the solution the faster you will see results....if you are impatient, you can simmer the solution on a hot plate using Pyrex (no metal pots) or suspend the glass jar in an ultrasonic cleaner. This solution is only meant for precious metals, and stones should never be put in the solution.