Hi ajohnson08099,
I think the foundation of your problem was taking the kitten away from her mother and siblings at 5 weeks of age ... which is too young. I'm just a simple cat owner but have some friends that breed pedigree cats. They never let their kittens go to a new home prior to 12 weeks ... and most wait until at least 16 weeks (give the last kitten series vaccination at 16 weeks, then wait 7-10 days before the kitten goes to a new home). My current kitten (now 2 yrs old) was 22.5 weeks old when I got her.
Those first few months are very important for a kitten to learn social skills. One of the important lessons is how to play and that biting hurts. Normally, kittens will play with their siblings and they will teach each other when to stop and when biting hurts. But a kitten that goes to a new home at 5 weeks hasn't had the chance to learn the lessons from its siblings and mother about biting and rough playing. It will be more difficult for you to teach her now.
Having the dog as a playmate isn't likely to help since she needed to learn the lessons from other cats. Cats still have a lot of wild instincts, and dogs are a natural predator. Cats and dogs can often become friends ... but a cat's natural instinct is to either hide or attack a dog as a potential predator that might hurt the cat.
I think most cats will get better about biting as they get older. It is always important not to play with the kitten using your hands as a toy. If she wants to bite something in play, give her a toy that she can chew on (be sure there are no small parts that she can chew off and eat ... which could cause a life threatening intestinal blockage). I've heard a number of suggestions on how to handle it. Some suggest, if the kitten bites your hand, not to pull your hand away quickly (which encourages the biting) but to hold it still and calmly put the kitten down. Others suggest saying a loud "ouch" to try to get the kitten to understand that it hurts (though that doesn't always work). Many suggest doing a "time out" ... when the kitten starts biting/scratching, set her down and pay no attention to her for a couple minutes ... or, if that doesn't work, put her in the bathroom for 5-10 minutes to calm down. The idea is to not reinforce the bad behavior and for the kitten to learn that if she bites then she won't get the attention she wants.
I think that once a kitten starts biting at a young age without learning about rough play from siblings or its mother, it can be a real challenge to stop the habit. But it is important never to get angry with her or use any force. She won't understand and she may just bite more. Cats aren't dogs and don't understand forceful discipline. Always be gentle and loving with her and try to redirect her behavior to a toy, or ignore her for a couple minutes. Hopefully she will grow out of it.
I agree with not using drugs. However, you could try getting some Feliway (it comes as a spray or a plug-in diffuser ... the diffuser may be better). Feliway isn't a drug, but rather simulates cat pheromones ... that helps the cat to relax. Many vets and specialists recommend Feliway when cats are stressed and/or have behavioral issues. You can check with your vet about it. It isn't a drug and to my knowledge has no negative side effects.
Good luck.
Stephanie in Montreal