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I am sorry to hear about the problems you are experiencing with your daughter. Issues related to eating are very serious and definitely require professional help. Your daughter is 13 and does not get the right to make medical decisions for herself. It is important for you to let her know that going to therapy is non negotiable. This is something that you will require her to do just like going to school. I would not suggest paying her for this, but requiring her to do it just like you would require her to get her medical checkups and vaccinations, come home on time, and brush her teeth every day. Your firm attitude about this will be important. If she tries to refuse, then you can calmly let her know that when she cooperates with this she will have privileges, but until then she will not. This could include electronics, and/or other privileges that go beyond the basics such as food, water and shelter. Remember not to fight about this and try not to show emotion about this, just be very firm and matter of fact that this is something that she will do; she doesn't have to like it, but she will do it.
Be sure to find a psychologist who specializes in eating disorders. Overall, individual psychotherapy for eating disorders uses a combination of cognitive-behavioral strategies and psychodynamic exploration to address behaviors and reduce symptoms and to examine underlying conflicts and issues, including the feelings that fuel the disorder. It is important to keep in mind that eating disorders are not actually about food, but rather about the feelings that food is used to help regulate and manage, and they are not just about body image or external appearance, but rather about how the individual experiences and feels about him/her self on the inside. Cutting is generally about expressing emotional pain.
DBT or Dialectical Behavior Therapy, has been shown to be helpful for eating disorders as well as cutting.
Your daughter is expressing her emotional pain through these behaviors, as well as her school issues. She needs professional help for this. You have every right to require her to go to therapy. Believe in this as you speak to her about it so that you will be able to be strong in letting her know that you are requiring this because you love her and care about her.
I hope this answer is helpful. Please let me know if I can clarify further.