
The pup sees and is treating the kids as litter mates. The mad dog bit is acitally over excitement and delight at seeing them.
How do the kids interact hith the pup. Do they physically play with it? When I had pups and kids were small no playng between pups and kids was allowed. By this I mean a dogs idea of play withich is rough and tumble, especially a staff as they like to play hard. Pups had to learn that they only go human attention if they were quietly in a down.
I would sugggets having a lead on the pup when kids get home.
They should be instructed not to make eye contact with the pup, and you can control pup with the lead with just a command of off or leave.
Any sign of the pup calming down should be praised genlty. Also strokiung very slowly when petting is calming as well as speaking quietly (kids don't they have excited hig pitched voices).
Also Staffs are a vocal breed, as in back chatting snuffling growly talking etc, it is part of their charm and doesn't mean anything much other than being excited or pleased.
Once pup has calmed down allow her to gret the kids but again they should only interact when she is calm. Stroke her very gently and slowly, instantly get up and turn away if she gets high. You can step on the end of the lead or hold it to ensure she doesn't chase them.
They should never run around in her presense. In other words she cannot be alllowed anything but supervised pats until she has self control, and learns that the kids are not littermates or exciting squeaky toys.
I have never found the squeeling method for bite inhibition ever works with anythign but very sensitive pups, for those who love to play rough just makes the game better.
In my own expereince ignoring or slight aversion with slightly older more stubborn pups has worked. For my breed a jet of water in the face to shcok them momentarily long enough to get a command in works well (I use a well rinsed frontline bottle set to jet, not mist), but staffs may find the water fun.