Firstly, huge hugs..
Having been through this 6 months ago I know how your feeling... The battle between head n heart is horrible. We had to put our 8yr old boy down after he suffered a stroke from a Tumor and didn't respond to the meds and it was one of the hardest things I have ever done.
Our boy was already at the vets due to them trying to treat the stroke so had already been hooked up to a drip, once we made the decision it was all pretty quick, our vet brought him in wrapped in his blankets, gave him a needle and while we were patting/hugging him he just slipped away to sleep.
I would leave the decision to the kids, but be prepared for the grief that comes with losing a pet, for me it was overwhelming.
Take care, whatever you choose to do know that it is the right choice for you & your puppy.