Hello, I commend you for adopting a child of a different race, that is beautiful and there will definitely be challenges along the way. I first would like you to seriously consider changing hair dressers, because any professional stylist that would put a relaxer in a child's hair that young DOES NOT have the health of your child's hair as their best interest.
Second, the term "Good Hair" is not the best term to use, it refers to curly hair as if non-curly hair is not good. Good hair is healthy hair and now that your child has a relaxer, if it is not maintained well it will no longer be healthy and no longer be "good hair". In the future a better term to use might be a curlier texture or finer grade, but "good hair" is not appropriate to describe a particular texture.
Finally, I would suggest seriously growing out the relaxer and gradually cutting off the relaxed ends and finding someone into natural hair care and getting your daughter's hair braided or twisted (not tight, nor with extensions) and wearing it braided or twisted regularly. And on weekends you can give her scalp a chance to breath and take the braids/twist out and wear puffs or pony tails (loose). As for products, there are several in target, walmart or sally's that will be great for daily use and won't dry the hair out. becareful of product buildup.