C.A.
I too have searched all over for sick child care. The only thing I managed to find is
Coughs and Cuddles at Presbyterian hospitals.
http://www.kiddieacademyofplano.com/SickChild.htm
I have never used it although it's in the back of my mind, the rates are also not too bad.
You might want to look into full time private home care, perhaps someone that will watch your daughter only with no other kids? That would be best for times when she's sick. I had an acquaintance that did this for me for a while when my daughter was tiny, but she's full time with another family now. That would probably also you be the most flexible for you from the hours perspective. Good luck, do let us know what you find out!