Hey there m2m2m!
It sounds like you deleted a few pictures of your kids' iPhone and you went to see if you could find the album for deleted photos, but you weren't able to find it - nor restore those old photos. And I can understand how you would've wanted and imagined that a process such as recovering a photo should've been so simple.
Kudos to you for trying to reset all of your settings on the iOS device. That's something I've would tried suggesting as well. 🙂
If I'm correct - Apple didn't implement the feature of making photos that users wished to be deleted, to being moved to a deleted album, before they were permanently deleted on iOS 7. This feature wasn't recently introduced until I believe somewhere around iOS 9 or iOS 10. Though I could be incorrect about this, but it may help to explain why you're not able to recover those photos.
But, you still may be in luck. Try checking out this knowledge base support article written by Apple here:
Get help with missing photos - Apple Support.
And just as a little reminder, if you happened to have any third party apps on the device - such as Google Photos, or Google Drive, maybe even Dropbox. Check to see if your photos happened to sync to those services.
Hopefully you're able to get your photos back, and if not - at the very least I hope I was able to give you some information, so hopefully something like that won't happen in the future. And some things that you can do to help back up your photos.
Take care for now!
😀
