This is probably "Don't Ask Jack", indeed by Neil Gaiman.
> This is an extremely eerie short story about a jack in the box who lives in an old attic in a creaky old house. The children who live in the house have been deeply effected by Jack their experience with Jack who are told things by jack that they "can't quite remember and can't quite forget".
Found by searching for _Gaiman "Jack in a box"_
## A reading of the story on Youtube
After listening to the video, it fits all of your details including the child who died in the war, the one declared insane after he was found in the basement with fire-starting materials, the nursery full of rats and the statement that Jack "could wait forever" for the children.