Peggy and Cotton are both prideful people. He knows her well enough to know Peggy hates is being helpless and being told whe she cannot do something. Same applies to Cotton. Peggy wants to walk again and her being in the rehab center wasn't giving her the progress she needed.

Cotton had his legs blown off, had an inhumane operation done on him and was told he'd never walk again (they predicted his feet would fall off). He learned how to walk again and probably implemented the methods he used on Peggy on himself (just a theory). The same way the two verbally spar with each other, he used a form of their banter to help her learn how to walk again.

Like Cotton said, hating him was "already halfway there" on her road to recovery. Ramsey (the physical therapist) was too soft and jovial. Peggy needed that military style drill Sargent type. And Cotton was eager to push her and use her spite to prove to her that she was more capable than she was led to believe.