Some of you city people.
This is alive in pain, and there is living. The two are worlds apart, and if something is in constant pain, it can not be resolve so the animal can have a normal life again so it not in pain/dying of pain long term, then putting it down quickly is the right thing to do.
Most of use will just put the animal down out of sight of the other folk via well placed shot to the brain to make it quick death, but if such is not in your wheel house, then maybe it time to take the animal to the vet to be put down EBI method instead. With EBI, your daughter to hold the cat at is expires, but some times the effect on the child if not metal prepared, can leave some deep emotion scares. Hence better to just sell the story that cat went to live on a farm out in the country, isntead of letting the child know you did put it down.
And before someone gets on a tangent, there is EBI method, or what a few of the animal shelters do, and that it vacuum chamber isntead.
Between vacuum chamber and gun shot to the head with enough gun, If someone was easing me onto the next plain of existence, head shot with enough gun to make is quick, verse the mins of suffocation before you would pass. Hell, EBI for animial is not bad since they are clueless on that is going on, but as human that way in regards to death death penalty execution, you know what is going on, so still a good min of panic before you do go unconscious, and God only knows what is still going through your mind until you do expire isntead.