Am very sorry to read about your kitten ... I've had many cats PTS over the years, and most for kidney failure. These were however all at an age where kidney problems weren't unexpected and had long and happy lives.
Please don't blame yourself. It's so easy when your pet is ill to think you should have done this or should have done that - I've been through that myself .... for example, I lost a cat a couple of years ago following a routine and regular procedure she'd have every 3 or 4 months, but on this occasion, the vet accidentally pierced an artery and she lost a lot of blood, had to be revived etc., and only lasted another week despite the vet's best attention in the meantime. It was an awful accident and I really don't bear a grudge against the vet but kept thinking that if I hadn't taken her that time, she'd have been okay. Thing is though, if I hadn't taken her, her quality of life would have been compromised (she had a non malignant but large cyst on her shoulder that needed to be drained or else it'd impede her movement). And the same is true for you - you know that having her spayed was the responsible thing to do and couldn't possibly have predicted her falling ill. I'm not a vet, but anaesthetic wouldn't usually kick start this sort of problem so far as I know - unless there was an existing underlying issue. You did all you could and it would have been wrong not to get her spayed.
(By the way, I spent that week trying to feed my cat which was heartbreaking - the vet suggested liquidising warmed food and syringing it into her mouth, which was traumatic for both of us).
You've all had a terrible shock and it's so hard to know what to say to or do for kids when their pet dies. In my case, my daughter seems to find it helpful to remember them by choosing her favourite photos and putting them in a "special" frame - the sort that has space for 4 or 5 - which she has on her bedside table and says "goodnight" to. Rather than them completely "disappearing" so to speak. In time, you might feel able to provide another cat with a home ...... there's no rush of course (I'd wait at least a few months myself), but being a lifelong cat lover, even when I've felt distraught at losing a beloved pet, some bit of me knew that sooner or later we'd get another - not as a replacement, but having the responsibility and the distraction of a new pet does help make the loss a bit easier, and that's seemed to be particularly true for my daughter each time. She'll still talk very fondly about past pets but the natural excitement of a new one definitely seems to help.