I skipped a grade in elementary school, but am not so keen on accelerating my kids for a few reasons:
1. I was still bored after the skip. It's the pace of the instruction that wasn't right, and being at a grade or two higher didn't change that.
2. It felt like I raced through the younger grades, where there was more individual attention from teachers, less negative attention for being "too smart". Grades 4-8 were the worst for me - sooo boring, all I really learned was how to keep my hand down and not call attention to myself. (Since your dd is so social, make sure she doesn't learn to play dumb just to fit in like I did.)
3. minor things that seemed important at the time: last to reach puberty/need to wear a bra
, graduating from high school without my drivers' license, graduating from college before being able to get into a bar with my friends, being sent on business trips before I was old enough to rent a car (but I'm sure being one of the older kids in class has its own set of challenges too)
4. Like shelbean91's sis, I don't think I was ready for college when I got there. While I did OK, I chose my major/classes for the wrong reasons. Had I been more mature when I was in college, I think I would have used the time so much more wisely. (Now I have a mechanical engineering degree that I have no desire to EVER use.
My kids have september and november birthdays, so I'm feeling a bit of the same pressure to start them early. But so far I'm resisting. I want to try to let them enjoy being little for as long as I can, and help them to maintain their curiosity and self-motivation. My eldest is 4.5 and in a montessori preschool, a great fit for him so far. I hope to put him in public school when the time comes, but am open to homeschooling if we find it doesn't work for him.
Instead of thinking about skipping kindergarten, I would treasure an extra year of half-day school and the time it would give you to explore the world together. From reading your original post, she's obviously learning a lot being with you. If she needs to skip, why not let it be one of the mind-numbing elementary school grades, not kindergarten which has more fun and movement and less busy work.
Enjoy your time together, she sounds like an amazing kid!