is your daughter's issue really careless mistakes, or is there an underlying comprehension problem. and what does "really good at math" really mean? as a high school math teacher, i always found parental descriptions of what their kids know of certain subjects can be vague. i'm not trying to be mean, i just want you to assess what's going on with a little more detail, more for yourself.
if you decide to go with a tutoring center, the most useful part will be a diagnostic test they can give your daughter to give you a sense of what skills are weak or strong. personally, i don't like kumon. i think they stress drill and kill too much. there is value in drill, but it doesn't work for everyone.
if you can find a tutor that has experience with teaching mental math skills, that might be more useful to your daughter. also problem-solving and reading comprehension. "advanced math" for the elementary grades is mostly making sense of words and connecting numbers to them. the mental math will help with comprehending things like double digit multiplication, etc.
by mental math, i mean teaching regrouping skills like
15-9 = 6, right?
it's also 15-10+1 = 6. 15-10 is a lot easier and then if you add back 1, you get the final answer.
those sorts of understandings get at larger math ideas that are very subtle but extremely useful for mental agility.
i used to tutor myself, but i'm due in 3.5 weeks, so i'm off. :oP
as for hesitancy on your daughter's part, i would stop by a teaching supply store, like morrison's school supplies in sunnyvale and pick up a math based game. if you help your daughter see that math has fun applications and that there are different ways to get the "right answer", she'll start to let go a little.
growing up, i played a lot of logic and math based games too. these days, they have sudoku and kakuro, which are great. perhaps the best thing they taught me is that struggle, making mistakes, and trying a different way is all part of problem solving. the students i see who "hit a wall" some time between middle school and freshman year of high school never really learned how to persevere through math problems, are terrified of not finding the "right answer" in one step, and get really confused when they are presented with 3 different ways to solve the same problem.
obviously, math is my thing, but i also agree...there are other things to enjoy! at the very least, let your daughter enjoy what she likes best at the same time. don't take it away, or she'll resent math.