I am a 53 year old male that has suffered 51 years of suicidal depression. I'm going to try and give you some insight into the suicidal mindset.
For 51 years, I was part of a dysfunctional family that ostracized and belittled me because they made a habit of taking out their own frustrations on the most gentle and kind member of the family,...ME. Anyway, I knew before I was 10 yrs. old that my parents were not a "match made in heaven" (they never should have gotten married). They separated when I was 15 and divorced years later, but it was when I was 10 that I (my mind) technically lost my right of existence (I should never have been born because my parents never should have married). I didn't realize that at the time, only my mind did. It couldn't figure out how it was that I do exist (oh, am I not making sense?...welcome to the Twilight Zone of the mind). I always lived as though I "existed" only and could not actually EXPERIENCE "life" (SELF/identity). I would do amazingly stupid, silly, and detrimental things (not cutting) and almost enjoy pain periodically as an adventure (recognition of "SELF", I didn't think of it as "PAIN", it was "SENSATION").
The cutting makes her feel alive (her mind can then acknowledge that she is ALIVE, not simply EXISTING). Don't explain these things to her, she will feel self-conscious and ashamed as though she is "demented".
I didn't consciously indulge in feeling sensation, but neither did I consider "sensation" something to be avoided (avoidance of sensation would be "sensory deprivation" of SELF.
All of my life I had a feeling that something was wrong (it was just a "feeling", probably something "in my life" or "about my life" is wrong). When I was 51, I heard someone (a "man" of the household) bragging about how he was "surviving" his divorce and I wondered how his two little daughters are ging to be reacting to the divorce. I never had that "feeling" again (it's the MYSTERY that hurt) because I had finally realized what that feeling was...my mind couldn't figure out how it was that I exist. Other people are happy having a strong ethnic heritage and a solid family lineage, but I had neither (technically, I should never have been born,..."man without a country").
Your daughter's bulemia gives her "sensation" (weakens her body, but that is "SENSATION"). The alcohol is more "sensation" from the tastebuds to the blood. Her body and mind panic because they are too weak to handle stress. Her intestines are too weak and go into shock when it comes time to digest solid food. Try vegetable juices or easily digestible food (sardines are very nutritious, cottage cheese, etc.). She feels alienation. Maybe it's because she has been recognized more for her ACHEIVED status from being in the 99th percentile (academic acheivement), not for her creative, artistic SELF.
Conscience and a "consciousness" are very important in order to realize "life". A sensitive, insightful, creative human being with "heart" NEEDS this and cannot truly be a "human" without it (will find life intolerable).
She has become ACCUSTOMED to not having nutrition (this enables her sensory deprivation). Nutrition would be a big help to her and she probably is NOT addicted to alcohol (PROBABLY NOT an alcoholic).