awww Im so sorry to hear about your babys health problems. I know that 8 weeks is a bit early for your baby to leave mama, but it used to be that way all the time, pups would go to homes between 6 - 8 weeks very often. Its really not your fault. 8 weeks is young, but not sooo young that the pups life is in danger. Please don't blame yourself.
A lot of pups lately have had diarrhea and throwing up. I just went through it with BOTH of mine about 3 weeks apart from each other.
I hope everything works out for you and she gets well quickly! I know how heartbreaking and stressful it is to be worried about your sick pup for so long. Hang in there!