I might be inclined to use the child going to the hospital as a reason to give out a health reminder. Asking because of the fact the child has asthma and coughing to the point she had to visit the hospital (leave out the part that it was because parents weren't treating the asthma) ask that parents please be as vigilent as possible in seeking a medical diagnosis of illness - especially coughing and other cold-like symptoms so you can limit the illnesses going through the daycare.
Then if mom doesn't take the child you would have every right to say re: notice - please get doctor's confirmation of why the illness.