Growing up on a ranch I think my attitude and experience with this might be different from the majority of people who post here. For instance you get exposed to death at a very young age. From finding the still body of your fathers faithful cow dog who was older than you. To finding the still body of a sick orphaned calf that you have been bottle feeding for a few weeks.
I think the most important thing to consider is what is best for the animal. It can be hard to let go and it can be hard to break your kids heart. But if you really love your pet, do the right thing and end its suffering. (considering that is the case, not judging the OP since I'm not a 100% familiar with his dogs condition).
When I was a teenager my mother had a dog that she had had before I was born. He was deaf and almost blind, but was still getting around. My mother accidentally drove over him because he couldn't hear or see the truck and bolted out in front of her unexpectedly when she was driving into the yard. Not only did she feel bad for killing her beloved pet. She felt guilty for not having him put down humanly a few months earlier.
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