First of all, my heart goes out to her about her family troubles but I have no advice. I don’t know what kind of resources you have to be able to help her or if it’s even possible, but I can advise you about the teaching aspect.
In my experience (I’ve got quite a bit) if someone is actively trying to learn about Judaism, they want to learn about Judaism. Simply talk about anything that she isn’t mocking at the moment and work back around to that later. Someone can’t stand hearing about yet another halacha, talk about dveikus. She feels she can’t relate to the concept of moshiach, talk about the shabbos table. She doesn’t believe in God — this will be very shocking — nu? Lots of frum people don’t. Talk about something else. I repeat, if she’s there, she wants to be. Keep going until you find something she does want to learn about, and since all of torah is tightly connected you can work your way back when you feel she’ll be receptive.