I think Arcadia in Whiterun has cough drops for sale. In all seriousness, I haven't seen this one before. Not sure what your options are.....
Probably only reloading a save from before it started, and hopefully OP has that save.
See if you can find a http://en.uesp.net/wiki/Skyrim:Potions#Cure_Poison potion and if that will stop your character's coughing.