I've had cats since the day I was born, except for a short while just after we married. The DH gifted me with the best Christmas present ever, my precious calico kitten 21 years ago. A year and a half later, we got a second kitten. I am allergic to almost everything on earth and much of it triggers my asthma. I take one or more antihistamines on a daily basis, use inhalers daily, often need a nebulizer, steroids, etc. It is my lot in life.
At some point in that 21 years of "married life with cats," I became allergic to cats. It just showed up on allergy testing when it never had before. I thought, "Oh crap....How did that happen?"

The allergist asked me if I would consider getting rid of the cats and I informed him I'd get rid of
him before I got rid of those cats. He said that most of his patients give similar answers. He pleaded with me to keep them out of my bedroom, but I said that might be difficult, as one of them like to wrap herself around my head like earmuffs.

I figured I had to take antihistamines anyway...What was one more trigger when I loved those kitties so much?
Of course I wondered what I'd do if we had an allergic child. We wound up adopting, but if we'd had a bio child, the odds were that the poor kid would have been allergic to squillions of things anyway, so antihistamines would have been in that's kid's future. As it turns out, DD's lactose intolerance is her only issue. No cat allergy.
FF many years and the first kitty died about 5 years ago. Just a few months ago, our last kitty died at 19 and I have to say it nearly killed me. This house feels empty without a cat. I need one. My allergic self needs one.

Yes, I am an adult and this is your child we're talking about. But, if she is mildly allergic and (like me) has to be on antihistamines for other stuff anyway, then removing the cat from your home is not going to markedly decrease her allergies. And it may make her feel guilty, even though she shouldn't.
I say work on decreasing the effect the cat can have on her and see how that helps. If she was having asthma attacks from the cat being near, it would be another story. But posters have given you good ideas. Get rid of carpet, keep the cat out of her room, bathe or at least wipe down the cat, etc. Tell her she can't snuggle her face in the cat anymore.
Good luck!