FMLA law and disability is tricky, I don't know how it works as being a care taker of a child with health issues, I honestly don't think she would be qualified as disabled as her asthma does not prevent her from experiencing life aside from her precautions and medications. Sorry, I've just seen families with such severe health issues of children and spouses so much worse than this and aside from tax breaks on the medical care, the parents didn't get any other support. Proving total disability and severe functional limitations is very difficult and it doesn't really sound like she would apply as being disabled anyways, though consulting her physician would be helpful and necessary for you to provide medical proof.
My sil had the same issues as your daughter as a child, anaphalctic shock, had to use the epipen numerous times, also in and out of the hospital... but her health progressed as she maintained her health and as an adult, she is perfectly fine and has her asthma and allergies under control, though still has to have the epipen around just in case (though she hasn't needed it in many years).
According to the law, "A child under 18 is considered disabled if his or her physical or mental condition is so severe that it results in marked and severe functional limitations. The condition must last or be expected to last at least 12 months or be expected to result in the child's death. And of course, the child must not be working at a job that we consider to be substantial work. "
Information on it can be found here:
http://www.ssa.gov/kids/parent6.htm
and how to apply and a qualification evaluation here:
http://www.ssa.gov/onlineservices/
As far as being a single mother is concerned, there are resources to help with that such as food/shelter..., and perhaps laws for some jobs to require time off for emergencies if taking care of a relative with a health condition, but not sure if they are really effective or if they really even exist for all levels of employment.
I hope your daughter can get well very soon and your lives will stabilize soon as well.
- just read your update, I would imagine that being on oxygen most of the day is very disabling. I truly hope you do get some sort of help. do you know what she is allergic too? I know humidity in fl is quite bad and can cause a lot more severe allergy problems. there are ways to help with that in the home, . to remove drapes, remove carpet (or at least lay down seagrass rugs), excess pillows... I read an article on creating an allergin home, lots of interesting ideas, I'll see if I can find it and post it.