I understand how this is frustrating. I have been in a similar situation with a cousin of mine just a bit younger than me that I consider my younger sister. I think what is important, for a person you care a lot about, is to let them know you're there to listen whenever they want to talk. Being persistent in trying to make small talk (if your cousin is not comfortable opening up about deeper issues yet) every now and then will help her feel more comfortable to open up later. With this said, I agree with what you said about making sure you do not alienate her by trying too hard. Give it time, and her reciprocal care for you will show sooner or later. Don't let if affect you too much...