Deciding between homeschooling and private school involves a lot of factors. Among them...your local resources, your willingness to devote a full-time schedule to this new job, your daughter's willingness to take this type of direction and independent learning from you, other children, social outlets, curriculum choices, financial considerations, etc etc.
In our personal case, we feel that we have to homeschool because there are no other suitable options for our son. We've tried public school, and the top private schools still can't give him the challenge he needs. We're lucky to live in a community where hs'ing is mainstream, though, and there are dozens of resources for us, along with more than 100 families we now know who are doing this along side us. We have a charter that supports us and pays for all the curriculum (that I choose). I'm also one of those moms who loves my new role, devoting every waking hour to either planning, teaching, or driving the boys to some activity with other kids/moms. Without any one of these factors, I'm not sure that it would be the success that it is for us.
I think the first thing you might want to do is check all your local homeschool resources (charters? co-ops? homeschool meetups or support groups?). Try to map out what you would truly need to make this successful, and make sure it's all available to you. IMO, even introverted children (not to mention Moms!!) need to get out of the house with other kids at least 1ce-2ce/week, so make sure you have the ability to do that.
On the other hand, if you have a gut feeling and you're both ready to jump in, I say go for it! The great thing about either decision is that you can change your mind at any time and try the other.
