I'm afraid I can't make any sense of "her drawing" either.
"A lead" means the same here as it does in the sense of a criminal investigation. It's similar to a clue. Her "secret problem" was that she was worried about being too big-boned to be a ballet dancer and thereby concerned that she was letting down her parents. Once that "secret problem" was out in the open, the psychologist could base his treatment on the actual problem and she found herself able to walk again.
Student or Learner