This appears to be a rather new relationship in which there is not only low trust but also an imbalance of emotional and informational self-sharing. Perhaps she has quite a bit of emotional baggage that she is carrying from past relationships or maybe from growing up (e.g., parents who went through an ugly divorce). People carry their pain with them.
Maybe she's emotionally guarded for good reasons that have nothing to do with you personally. Is it worth trying to work through to see if that's true? If so, rather than jumping to anger, first try a more open, playful approach, such as 20-questions or an "ask me anything." Mutually establish the ground rules up front before anyone asks any questions.
For example, here are some sample ground rules:
1) On individual slips of paper, each partner writes down ten questions that they want to know about the other partner. It has to be a question that applies to both partners, not just one partner. The question can be philosophical, about past experiences or relationships, a silly question, practical question, a "what would you do scenario" or a question about life priorities, dreams, or values. If there are any off-limit subjects, agree on them in advance. Fold the slips of paper and put them in a jar.
2) Over pizza or in a relaxed setting, the first partner draws a random question from the jar and answers it. The second partner simply listens, and they can ask follow-up questions for further clarification. Then the second partner answers the same question, reversing roles.
3) If either partner comes to a question that they truly don't want to answer, they shouldn't be forced to do so. If possible, they might offer why they don't feel comfortable answering. At least now you'll know what the sensitive subjects are.
You might start out with lighter, very emotionally non-threatening topics for the first game or two of 20 questions (silly questions, favorite song, what would you do scenarios) then gradually sprinkle in some more emotionally revealing questions.