From your well-thought-out letter it is very clear how much you care for your friend. But the truth is you are not going to be the one that can change her. If it really is APD the heavy lifting on that is going to be done by a professional. But there are some things you can do to help.
The dynamic of you helping and trying to figure out how to change her and make her better is actually part of the pattern that needs to be broken. Here is the cycle: You try, she fails, you feel frustrated, and she doesn’t get better while you wrack your brains trying to make her feel better. What I would encourage is for you to stop this cycle of trying to get her to feel better. From your description nothing you have done has worked, and if makes you feel frustrated it is time to stop.
Instead I would find ways to accept your friend as she is, and highlight the genuine things she does that are positive in spite of her emotional and behavioral patterns. This isn’t to fix her, per se, but to remind her that even though she has some avoidant patterns she still is able to have these positive features.
Finally I would tell her what your frustration and difficulty has been, and is, being around her. If this is going to be an honest friendship then you saying your truth, and her being able to hear it, is important. You are not her therapist; you are her friend. Friends need to get something from each other and give something to each other. Let her know how you feel and what you need. Otherwise you will spend your time trying to rescue her, and being frustrated.