How is Madeline's problem fixed? What was the cause of her disorder in the first place?

Madeline's problem is fixed when Dr. Sachs tricks her into making use of her hands. Apparently she had always been able to use them but perhaps from babying by her family she came to believe that she could not. The nurses simply place a plate of food out of reach from Madeline but only slightly, so that she had reason to reach out further for the plate than usual. As a result, she discovers that she has use of her hands. She then decides to take up sculpting.