Hi there!
I'm sorry I can't help with information on the animal rights aspect of things. However, I went through a drawn-out illness with one of my cats with my 2-year-old daughter, and after almost 6 months of us trekking back and forth to the vet's, leaving him there for a week at a time, and having him sequestered frequently so I could monitor his food intake and urinary/feces output, he got sick enough that it was time to let him go.
What I did with my daughter was to be upfront without going into detail. When my daughter asked me, "Where's Linus?" I would reply where he was if he was at the vet's or sequestered in my master bathroom. I just would say that he was sick and the animal doctor was trying to make him feel better.
When we had to take him for his final visit, I was extremely torn apart - this cat had been my baby (and still was to an extent) until I had human children and he and I had a special bond. My daughter liked him a lot too. When I came out of the vet's office and came home without him, my daughter did ask, "Where's Linus?" and I just told her at the time that he had to stay at the vet's because we couldn't take care of him anymore - that he had gotten too sick. She'd ask about him off and on again for a couple of weeks or so, but it got dramatically less frequent and now, after about 6 weeks since he went to sleep, she doesn't mention him at all, nor does she seem overly traumatized by the event.
We do have other cats and a dog, so I don't know if that part of your household is similar to mine, but at the age of 13 months (even younger than my daughter), not only will your daughter probably not remember the dog for very long - if the worst does happen it shouldn't cause her long-lasting trauma.
So what I'd recommend, based on what worked for us, is to let her take the lead. If she seems like she's asking about the dog, tell her that the dog had to stay at the vet's (or that the dog wasn't able to come home), and leave it at that. Chances are, though, that you won't have to deal with questions for a very long time because for a child that young, emotional and intellectual development won't make her too worried yet about a family pet who isn't there.
I really hope that your Daisy pulls through, and I'm really sorry that your poor dog has to go through all this. I'm pulling for her and your family!
J.