First try to recover completely from the broken cartilage
as your doctor diagnosed because diagnosis based on the radiological tests are almost correct. Ruling out the cartilage damage, coughing up the blood with a history of smoking and if you have fever suggests a condition called Bronchitis
(could also be Bronchiectasis
depending on further diagnosis). Consult a pulmonologist
, he might prescribe some bronchodilators or specific treatment.
Stop smoking (at least TRY to stop it). You will find improvement within a week.
Good luck, Thank you.