What types of illnesses does art therapy work for?

Most of them but- Counseling and even medication are usually part of the treatment. It's more often used in children or in patients who have serious problems or trust issues and are not good at talk therapy or have so many fears that they avoid medication until they are more trusting of their therapists and therapy. Art therapy is useful in certain brain damaged or other neurological disorders as well.
Art Therapy. General illnesses such as flu, heart disease, cancer. Also for crisis interventions & bereaved children such as ones with history of traumas.
PTSD. (Post-traumatic Stress Disorder). Years ago, I consulted at a hospital program for sexual trauma. It was difficult to engage the adolescents in group but the expressive (art) therapist brought out their innermost feelings. I invited her to work with me in my PTSD practice and the results were excellent. IMO expressive therapy is an underutilized, cost effective therapy for many conditions.