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There is a good chance that this is contributing to your child's illness. Some people are highly allergic to pet dander and it sounds like your child is as well.
Petting the cat 6 months ago would not produce contact allergies today.
Your daughter would probably find great relief if you did pull up the carpets and installed new carpet or hardwood floors.
You will want to start in the room where she sleep and the living room. This will probably be the most advantagious to her.
I hope this has helped.
Let me know if you have more questions.