She's 11 weeks old. Jumping and biting at this age is like human babies grabbing earrings/necklaces/hair and yanking on them -- you have to practice a certain amount of self defense, and a 3 year old human isn't capable of that.
>And she has done this since she turned about 8 weeks old.
When did you get her? If she left her mom and siblings much before 8 weeks old this could explain a lot of the behavior (though most of it's probably perfectly normal).
If the bite inhibition is working with you, that's great, and shows she understands it. She doesn't quite appreciate that it transfers to the kids yet, and also probably views them as her littermates. This is why it's so important to always supervise children and dogs -- they misread each other and then there's trouble. Kids wave their arms around and screech and run and generally send out great big "come chase me!" flags. And to children, puppies look cute and sweet, and then they leap up and nip you on the chin.
My main advice would be immediate puppy kindergarten, to lay the framework for basic obedience and also start building your relationship. At the same time, I'd let the 6 year old demand that the puppy sit for treats, to go outside, etc. Once the pup knows "sit" and the child knows how to deliver it calmly and firmly, you can let the child exercise authority in this way (under supervision). You could even let the 6 year old put the food in the bowl and feed it (after the pup sits -- you may need to hold the leash at first while they work on "wait") Controlling food is a BIG sign of power, and will help the child gain authority. However, always supervise any interactions until the dog is much older and you are 100% comfortable.
I'd limit the 3 year old's play time with the pup to when the pup is already quite tired, or when you can be outside and the child won't be the focus of the puppy's energy.
The biggest thing I'd advise you to do is keep a mini diary for a couple of weeks of the times of day that the pup exhibits the out of control behavior -- chances are she does it just as she's ready to fall asleep. In my house we call it "getting nasty" -- when the biting and turbo running start, and it's been an hour or two since the last nap (especially in the early evening, when it's generally worst), we slip on the leash, go outside to potty, then back inside and STRAIGHT TO THE CRATE. No drama, no cooing, no treats. (If you have a fenced yard and usually send the pup out, take her on the leash anyway, rather than get into a chase outside when she doesn't want to come back in).
You may be surprised at how often she'll be asleep and snoring within a minute or two. (Keep the crate wherever you are -- living room, etc. If you keep one in the bedroom, consider buying/borrowing a small one for the next few weeks to have handy in the living area.)
At 11 weeks she should be sleeping much of the time still -- maybe 1-3 hours up, 1-3 hours asleep, finally conking out in the early evening until early morning (with a trip outside before you go to bed and maybe one in the middle of the night, depending on her bladder).
. And this morning when my 6 year old Clayton was getting ready for school, she just followed him all through the house, and actually hugged him and gave him lavish kisses. What happened to the demonic puppy? 