Putting them to sleep is the most humane way. One drug to sedate, one to stop the heart. They are all heartbreaking. Its bad enpugh when they have to put one down because they are too ill to save.
We had to do this last year to our perfectly healthy kitten who had somehow ingested dental floss and would have attacks where bowel pain would hit and she'd race around the house trying to get away from the pain, spraying diarrhea everywhere. These attacks became more frequent after she and her sister got spayed. Our only three options were make her an outside cat (with her sister living indoors) where the raccoons and coyotes live, put her down or do surgery which the vet said it was unlikely she'd survive, being underweight and weakened by her condition.
We still cry about it.