FP, First let me say that I am sorry for what your family has endured. I am not without compassion. I also understand what it's like to live with someone who has a mental illness. I have no doubt that many people who abuse substances are self medicating, in effect, due to some kind of chemical imbalance, and to be quite frank, certain mental illnesses, I suspect, are examples of what the Bible was talking about with regards to demonic possession. I want to clarify though, that all mental illnesses, are not the same, and shouldn't be lumped together as such.
Okay, first of all, what AA would consider to be an addiction, covers too many things that are not the same. What I mean is that people who are escaping from something painful, and people who just have a chemical imbalance, are not the same. The person who is escaping pain, can be set free when the pain is healed, by changing their patterns of thought. While not easy, it's doable, and frankly from my experience, it seems that there are many in that camp.
Secondly, those with a chemical imbalance can often receive proper medication, as opposed to self medication through the wrong sources. Now this will possibly sound crazy, but in many cases even diet can balance the chemical imbalance. My mom has had some kind of mental illness that was observable since she was about 38, or so. It's a lot like Alzheimer's. She acts like a toddler, and demands her way. She also often doesn't understand what people are saying. She will repeat things like a broken record, etc. Now,about a year ago, she had a stroke. After the stroke, she couldn't swallow, so for about a month or so, she was on a liquid diet, and was fed through a tube in her stomach. Now, food has always been her joy, with mealtimes being the joy of her day. So you would think that without mealtimes, she would have been miserable. But while she was on that diet, she was just as lucid as could be. She was happy, and understood things. This made me realize that a proper diet can do so much more than I ever realized. Therefore, seeking God for proper discernment is crucial. But in none of those cases would I say that it should be assumed that the person will always be afflicted. While not every person gets healed, it's important to note that in scripture, Jesus healed everyone in his path who asked, or who's loved ones asked Him to.