At 9 weeks, she is not housetrained (you are just getting your timings right), and she has no way of knowing how she should behave to illicit attention and play - so she behaves as a puppy will with other dogs/puppies - bouncing, jumping up, barking and biting.
This will be made worse if she is frustrated, not sleeping properly, over tired, under tired, when she starts teething etc etc.
Use a room divider to split your living room so she can be one side with you, kids the other side.
Work on training her to sit or give eye contact for attention/fuss.
Ensure she sleeps well by having her crate in your room or bed beside your bed, this will speed up toilet training and you'll have a much better idea of how she is coping with life (tough times = needy puppy, easy times = confident independent puppy) - their confidence comes and goes over the first year or so, so don't be in a rush to move her to sleeping by herself any time soon.
Once thats in place you can then let the kids be with her when you are there to supervise and manage both her and their behaviour, rewarding her for doing the right things (once you have taught them!), managing and redirecting and avoiding her doing the wrong things.
Explain to the children she is a baby, she does not know how to communicate properly and she won't for a while, they have to be patient and follow instructions.
Puppy first months are tough going and particularly for kids who almost certainly believed a puppy would be a cute cuddly friend straight away and are horrified at this barky jumpy bitey monster who frightens them and does not listen. You have to manage their expectations, as well as your puppies needs/behaviour/training.