Hello
You are having cough for 5 months and 
took cough syrup as well some antibiotics . You got some relief but cough still present , particularly when take milk , ice cream ( milk products ).
Such type of cough may be due to these possibilities , these include:
1 
Allergy to  dairy products , particularly , you mentioned that you develop cough after milk and ice cream . Diagnosis can be confirmed by  allergy test for milk products . So consult an allergy specialist and get allergy test.
2 The next possibility of such symptoms may be due to asthmatic allergic 
bronchitis ( in this condition patient develop cough due to 
bronchoconstriction caused by allergy by mold, dust, pollen, fumes , 
smoking ) So take care about these allergens.
3 Smokers cough , if you are smoker then this cough  may be due to smoking.Be careful about that also.
however , there are so  many other reasons of such cough but consult another physician and get his opinion as mentioned above.