Wow, sucks man. Had to do that a couple of years ago. Hold on to the good times. Remember they don't live much longer than 13 years anyway and focus on the positive. When I had to put Maggie Mae down, I told the kids she was very sick and a doctor was going to try to give her a shot to make her feel better but it may not work. They spent the rest of the day with her and the next morning had a very, very good vet friend come to my place and held her as he administered the shot (I felt I owed that to her as she hated the vets office). Absolutely the toughest, saddest thing I have ever had to do. But you can't keep them around if they are in pain.
The kids understood, they will be sad but let them say their good byes. It's important. The pain never goes away but focus on what a good life he had! :way: