At least she knows that rattling your door will only result in the cops being called, whether you are home or not. She is now on full alert that You set your alarm! This should serve as notice to her that you are not unprotected, and she will not be allowed to get away with such sillyness.

I would replace the doorknob and NOT charge her for it. The next time she visits, I would point out the shiney new knob and explain that it was broken in her rage to open the door, and while she is family, and you understand that she felt her desire to enter your home was dire, if it happens again, she will be sent a bill for the replacement, and since most police and fire departments are now charging folks for false alarms or alarms that result in family members being found at fault on scene, she will be sent the bills for the emergency response that is created by such panic. She clearly knows your cell phone numbers, and should think to CALL FIRST! before creating property damage. You are happy to come home and discuss whatever earth shattering event that could cause her to try to rip your door off the hinges, but AFTER a phone call by her, not a call from the alarm company. Everyone gets a ONE free pass, and she just used hers.

Its so hard to deal with family members that behave in bizarre ways. I have a family tree thats full of nuts. You are doing great by having an alarm, and having boundaries that you stick to. She wont see you as an easy target in the future.