Sorry to hear that the end is near for your dog. This past May I have experienced the same with my dog..she was 12 years old....10 weeks old when I brought her home. I love my dog, miss her...but know in my heart I did the right thing in putting her down. Her health was failing, she wasn't functioning, stopped playing and lost interest in everything and finally one morning, her eyes told me she's had enough. I knew then in my heart it was time to let go and what a painful reality that was....Anyway, I called the vet, a friend, and off to the Vets office we went that morning.....
Talk to your children...with the dog in the same room...let them hear from you that the dog is very old...very sick and his time to go to heaven is coming soon. Hug, pet the dog as you are talking and maybe the children will want to do the same. I have no idea what questions your children will ask but try to keep the answers simple and express your sadness that the dogs health is failing and when he does pass, how you will all miss him. It's ok to cry and embrace your children if they do.
I tend to think to make it easier on all...It's best not to get into the reality of what takes place at the Vets office...nor mention your plans of taking the dog to the Vet.
Pick a morning when your children are in school to take the dog to the vet. This will give you some hours alone to deal with your own feelings after the dog has passed. Go with a another adult, a friend, a family member for support....I was glad I did as I was so upset I wouldn't have been able to drive.
When the kids come home from school I am sure they will sense something is not right....sit them down and tell them the dog wasn't responding well that morning so you took him to the Vet...and the dog died while there. Mourn with them....
One of the things I did was I made a mourning corner on my coffee table. Where I placed a picture of my dog, her collar (which I brought home with me from the vets office))) along with a bit of shaved hair the vet cut for me on request((and placed in a sm. plastic bag))) her favorite toy and lit a candle.
Ask your children to join you...hold hands and say a prayer....cry....then begin to talk about the good times with your dog....tell a story...maybe each child would also like to tell a story as well...maybe not.
Don't be so fast in removing the dogs things from the house either....give this a few days. One of the important things about healing from the loss, is allowing the feeling of loss to happen. Both you and each child will feel it differently...each child will let go differently...give them time but slowly move each of them in the direction of moving forward and getting on with daily things....as you slowly do the same for yourself.