My bathroom A/C has been leaking for about a year and I have called my landlords about 10 to 15 times letting them know that the leak is also causing black mold to grow in my bathroom. Can I break my lease because they have not come to fix it and the one time they did they just put plaster on my ceiling because the water was making holes in my ceiling.

Put it in writing, first of all. Say that you are notifying the Health Dept on day X.
No, you can't just break a lease based on this. You can ask, or you can prove your case when/if it isn't fixed.
I always keep my bathroom door open when not in use, and ask people to leave it ajar when the shower is on, even though the exhaust fan is on. No towels kept in there either. Bathrooms often have moisture problems no matter what.