You're not going to want to hear this, but I've looked over your dropbox file and the professor has done everything he should have done in the situation from the university's perspective, including explaining quite early on that the grade was mainly due to the lateness of an assignment.

Additionally, while "sick" as in pneumonia may not merit official accommodations, but is worked out between professor and student, "sick" as in psychiatric diagnosis and on medication long-term should, but you have to apply for such accommodations. You cannot bring them up post facto, whether or not the department chair knows about them.