Okaaaaaay, so I’ve had this problem with other pieces with broken axles, and here’s my best advice:
You’re definitely not getting it out of there without potentially damaging the broken axle, so get yourself a pair of very pointy tweezers or a pin. You’re going to want to feel along the sides of the pin for a potential gripping point if the tweezers, or a small gap if the pin.
Tweezers can very slightly grip the sides of the axle and allow for some leverage on pulling out - it will slip 90% of the time so have patience, young one. The pin is much harder and much more time consuming, especially if you want your mata lightstalk to be scratchless.