I would say it's when Mattie's grandfather dies. At this point, she has to decide what she's going to do. Her life from before is gone, and at first she just wanders the streets of Philadelphia. When she find Nell, she finds a purpose in life. She joins Eliza in her fight, and Eliza allows her to become an adult. Mattie rebuilds the coffeehouse, takes care of Nell, and her mother when she returns. Her grandfather's death was the point where Mattie had to grow up and take responsibility for herself and others.
I would have to say that the climax is when Mattie's grandfather dies. At this time in the story Mattie must decide what she is going to do with her life. When she finds Nell she has a reason for living and continues to fight. She finds Eliza and helps take care of the sick and dying. When her grandfather passed away she was forced to mature and become responsible for not only her well being but for the well beings of the ones around her.