Was the dog purchase upon "advice" or the "recommendation" of a licensed medical professional?
Emotional support animal: Not a guide dog. This is a gray area, see below. Claim the deduction at your own risk.
Your child can be your dependent if she is over age 19 but is disabled, and does not provide more than half her own support. For tax purposes, disabled means unable to perform gainful work due to a condition that is permanent, or likely to last at least one year, or likely to lead to death.
If she is your dependent, you can deduct medical expenses, or pay for her medical expenses from an HRA, FSA or HSA. The deduction is hard to reach, you can only deduct the part of your medical expenses that are more than 10% of your income.
An emotional support animal is probably not deductible. Here's the problem: a service animal is an animal that is trained to perform one or more assistive tasks for a disabled person. The costs of buying, training, and providing room and board for a service animal, are allowable medical expenses. However, an emotional support animal, that provides comfort through it's own natural behaviors and has no special training, is a gray area. Emotional support animals don't provide "treatment for a disease or condition" if they aren't trained. Of course, some emotional support animals are trained -- such as to detect and respond to seizures or emotional outbursts. If the animal is trained to perform some kind of assistance, then it is an allowable medical expense. If it only provides comforts through it's natural untrained behaviors, this is a gray area and different authorities go both ways, as far as I know there is no definitive IRS ruling.
You can use the child and dependent care credit (or a dependent care FSA) if you pay for care for your dependent who is unable to care for themself, if the care is provided so that you (and your spouse if married) can work. But expenses for home services (cleaning, housekeeping) are not allowable unless they are a trivial or incidental part of the day care. If you have a home care aide who provides care for your child while you work and also cleans the house, etc., you will need to divide the cost.
A home care aide may also be an allowable medical expense if the aide provides medical care. Again, you have to split their time: an aide who provides medical care 25% of the time, housekeeping 25% of the time, and dependent care so you can work 50% of the time; you can apply 25% of the cost to a medical expense deduction, 50% to the child and dependent care credit, and no deduction for the time spent doing housekeeping.