Will medicare pay for a broken nose that's causing these problems?

If you have breathing obstruction from a septal deviation being secondary from trauma or congenital, the insurance should cover this.  If any cosmetic changes are done at the same time, these would be considered out of pocket expenses.  

Each plan is different and reimbursement from insurance is becoming increasingly difficult. Medicare should reimburse for septal deviation and other functional issues. However, the process of reimbursement is such that many surgeons are requesting payment prior to surgery.

Hi and thanks for posting this question!
You will have to verify this generalization with your individual insurance carrier:
Nasal surgery addressing the septum for reasons of breathing difficulty or to repair traumatic injury is generally covered by insurance companies.
Nasal surgery to improve the appearance of the nose is considered cosmetic and is an out-of-pocket cost to the patient.
The best way to find out if you meet the criteria for insurance coverage is to be seen and evaluated by a reputable, board certified specialist in nasal surgery (either plastic surgeon or ENT surgeon).

Best wishes