I'm so sorry about your cat. One of our dogs died about a year and a half ago, so I can sympathize. My ds was your dd's age when our sweet Hannah died. My dd was 7. They were with her when she died (in the car on the way to the emergency vet). I was out of town, so dh had no choice but to take them. We had talked to them about death, especially regarding the dogs, for most of their lives. Hannah had life-long health problems and our other dog Maggie is 16, so she's been pretty old for quite a while.
As for the cremation, we have told our kids for a long time that if dh or I were to die we would have our bodies "turned into ashes." They have been to open-casket funerals, and we have used these times to talk to them about choices for remains after the body dies. They never asked exactly how the body gets "turned into ashes." I would explain it if they did. You could tell your dd something similar to that.
For the record, the kids were very resilient when we lost Hannah. They both cried for about 10 minutes, then a few short cries later that day. The next day they were on petfinder.com looking for a new dog or just pictures of dogs that looked like Hannah. We did wait about a year before getting another dog.
I agree with the pps about just being honest with your dd about the cat's death. The pet crematory here is not associated with a vet's office. They do pick up animals on request. We requested that they pick Hannah up at the emergency vet. We didn't get a fancy box for her ashes, just a plain one. The ashes are in a ziploc bag inside the box. The kids still like to open the bag occasionally to "feel Hannah." So do I.
It will be hard for a while. My thoughts are with you and your family.