The way your reader fell, the stylus probably bent while/before breaking, and that means the reason it's stuck is because that top end is wedged against/into the side of the stylus slot, and the remainder of the stylus in there is possibly a bit curved, which will make it harder to get it out.
Patience, patience, patience, is key here. And a red-hot needle, and long thin tweezers. As for how to get the needle into the stuck stylus, here's how you get one into a plastic cork stuck in the neck of a perfume bottle:
1. Heat up needle in a candle flame. Do NOT hold it with your hand. Seriously. They tortured people like that in the Middle Ages. Hold it with tweezers. (The needle, not the candle flame.)
2. Take very hot needle, press tip gently against the top of the plastic. Maneuver the tweezers so that you're holding the needle steady from the side, leaving the not-so-red top easily accessible.
3. Tap top of needle over and over and over and over and over and over and over and over and over and over with a very small hammer, or narrow hard object that can be used as a hammer. The handle of a screwdriver works well.
4. Keep tapping until the needle has gone deep enough into the plastic that it's firmly stuck, then let go and leave the whole thing alone until it's fully cool and the softened plastic around the needle has hardened again.
5. Grab the needle and patiently coax the perfume bott--,er, stylus, out from here it was stuck. Take your time, and resist the urge to force things. Let it take the time it takes, and don't underestimate the importance of wiggling sideways.
Good luck.
