I had cats for many years, the last one passing about about 10 years ago. I had to "be the grownup" when it came to taking them to the vet at the end when they were ill and the quality of life was gone. It doubly breaks your heart to lose the cat and have your children so unhappy. It sounds like Lily went on her own terms. My pooch Poppy died in her sleep at home in her favorite spot in early 2019; also on her own terms. That's the best we can hope, is that they don't suffer through painful medical treatments and go when they're ready.