Long lasting since a month - dry cough.

GM sir/madam, My son is 3.4year old and started with simple running nose symptom and then converted into dry cough. It has been 1 month that he is on treatment. Initially he took cold related treatment for almost 2 week i.e cetrizine, phenylephrine, dextromethorphan, chlorpheniramine. but having very little improvement. Further he developed ear pain so he has been given with Augmentine antibiotics treatment. But after that he was relaxed for 2 days only and then again started with severe dry cough. His vitamin D is found only 53 and he is taking weekly 60000IU. Doctor told it's bronchitis so gave him.pump of Levolin and Budocort 3 times a day even after 20 days of pump treatment very little improvement along with pump. Further now he has been given with Prednisone and cefixime treatment for 5 days but still he is coughing continuously, today is third days for this treatment. But very less improvement. He doesn't have fever at all. Please suggest me, what I can do for his improvement.