When I first brought home Momo @ 8weeks old, he got hypoglycemia...thus his immune system was shot & got kennel cough. My vet said it could've already been in his system & the hypoglycemic attack brought it out. He was sick for 3 months. Countless visits to 3 different Vets later, the last one upped the dose of amoxi-drops to 5ml from 1ml. That finally worked...but...the 5ml of antibiotics was too much for his digestive system (he was only 14 ozs). Wish I knew back then about yogurt as a pro-biotic. He's ALL GOOD now(a whopping 4 lbs).
Hope your baby gets well soon. Oh, & please don't triple or quadruple his dose of anti-biotics (just because it worked for Momo)...if you're concerned (which I would be, 5 months is a long time, but his cough did reduce significantly), please see a different Vet. I think it's good to get different opinions.