DO NOT USE A DREMEL
DO NOT USE EXTRACTORS
There is one, and only ONE way to effectively remove the screws broken below grade.
First of all, the screw is in there because its TIGHT to begin with, so the extractor is not going to do much, as it took a full slot on the head to break it clean off instead of moving the threads.
Here's how to fix it.
Go down to the local hobby shop, but also, welding supply places have similar. Get some steel tubing that is "just bigger" than the body of the screw. Now take your dremel, or a hand file, and make some nice even, and sharp, V notches in the end of the tubing. Cut about an inch of the tubing (with the notches on the end) and place it in a crank drill. Now drill over the end of the broken screw - in the same direction you want to turn the screw. Go slow, light pressure and remove wood a bit at a time. Soon, the screw will come out, with a little wooden plug too. Now, go back down to the hobby shop get some small maple dowel, and plug the hole. When the glue is dry, redrill it. And try again. You'll not see "the repair" because the tuner and screw head cover it.
This is the only way... trust me trust me trust me.
My reference is Frank Ford: www.frets.com
I've used this method myself on several used guitars I've gotten, that had broken tuner screws. In every case, it worked flawlessly.