Moisture inevitably corrodes the metal parts around a kitchen or bathroom sink, including those inside the faucet. When screws become stuck, you can't disassemble the faucet for repair until you remove them. Whenever a screw becomes stuck, you can often lubricate it enough to turn it, but if the head breaks off, you have to resort to another strategy that involves a fitting for your drill called a screw extractor. It isn't expensive and is a handy addition to your toolbox.
Turn off the angle stop under the sink to shut off the water to the faucet. Spray lubricant around the shank of the broken screw and wait 5 minutes for it to penetrate.
Insert a bit that's smaller than the shank of the broken screw into an electric drill. For the screws holding most faucet handles, a 1/8-inch bit should be small enough. Place the head of the bit on the shank of the broken screw and, holding the drill steady, turn it on and drill a hole about 1/4 inch deep into the shank.
Open the faucet all the way by turning the handle counterclockwise as far as it will go. Replace the drill bit with a screw extractor, which is a conical tool with a reverse-thread. They come in various sizes, so choose one that has a head small enough to fit in the hole you drilled.
Set the drill in reverse and place the tip of the extractor in the hole. Start the drill at the slowest possible speed, pushing the head of the extractor into the screw as you do so. The extractor threads should catch on the metal while the power generated by the drill forces it to turn counterclockwise.
Drill a slightly larger hole if the screw doesn't turn and the extractor begins to slip. Try the extractor again. If the screw is still stuck, heat it with a propane torch to loosen the corrosion on the threads and try again.
Replace the bit with one that is the same diameter as the screw shank if you can't remove the screw with the extractor. Drill into the screw until you have drilled all the way through it, then set the drill in reverse and clean the shards of metal remaining on the threads by brushing the bit against them. Remove the bit and brush out the opening with a narrow wire brush.
Lubricate the opening with spray lubricant or oil before screwing in a new screw.
Wear safety glasses while drilling to protect your eyes from fine shards of metal.