A skull fracture is a very serious injury. There is an area of the brain in the brainstem that plays a major role in the cough reflex. It is called the cough center. It has nerves that bring signals to it and nerves that carry signals out of it. The nerves coming into it bring a message that a cough should occur, for example choking on a piece of food, and the nerves exiting from the center carry signals to the muscles that work together to produce a cough.
There is no easy answer to your question but it is likely that the head injury damaged one or more of these nerves and the cough reflex is no longer completely functional. Either signals are not being received from the center or signals are not being sent out of the center. This may or may not be reversible. Only time will tell.
As a sufferer of chronic coughing & problems with breathing when in bed, I'd just like to ask if you have considered the bedroom's environment as it relates to allergins to your little one? Most people don't think about pillows, mattresses, stuffed toys or books in a room as 'problems', but I learned with myself & my son both that these all plays a role in making us sick. Thus slowly reducing our ability to breathe while trying to sleep at night.
My allergy specialist suggested stripping the room of my son, bare of everything other than the bed & then taking the bed's mattress & putting a special case over it to reduce allergins from the mattress from bothering him while he was sleeping. Also did the same for his pillow. It works amazingly.