What is the most humane way to kill a dog?

Context: My mom has a dog. He's just 18 days old. He falls sick frequently. He has severe diarrhea. He vomits frequently. His fur is dirty and sticky. He's not eating food. We are taking him to the vet every day. His condition just isn't getting better. We have been spoiled by the experience of taking care of our old dog. Old dog is healthy. He was never sick in 5 years. This puppy is weak. Above all, he whimpers all the time. He just whimpers for no reason. His condition is not bettering. So, I asked my mom and mom said to take him to the vet and ask them to kill him. I think that's the most humane way to end his life. I just want him to be gone, just because I want peace in the house. It's so stressful seeing him so sick and weak.