Hi A.,
Most coughs and respiratory issues end up being viral in nature and have to go away themselves since antibiotics don't work on viruses. Coughs that seems to linger for weeks at a time indicate viruses as well. However, most pediatricians want to see their patients if a cough has lasted longer than 2 weeks, just to make sure nothing else is going on. If there are no other symptoms, such as fever, difficulty breathing, coughing up green or yellow phlegm, it's just a wait it out diagnosis. But my recommendation is to be seen by the ped just to confirm. Lots of fluids and be sure not to share any eating utensils, drinking cups, etc. Clean everything that goes in your child's mouth with disinfectant. These things generally last so long because children keep reinfecting themselves. Be sure to have your child stay out of your 4 month old's face, too. You certainly don't want that passed on to your baby. And my other recommendation is to change toothbrushes frequently for a while. Any over the counter child cough medicine may help during the night, but it's better to stay away from them during the day if you can. Generally the cough is more irritating than dangerous. Hopefully it clears up soon. I feel for you! God bless.
-J.