Trinania,
I second spgirl. She knows how to read- is it level 1, 4-5 or chapter books? You will be surprised to know that many kinders know how to read already but some do not know A to Z. It depends a lot on the parental involvement and there is a huge knowledge gap between kinder kids. (in my DD's class in 1st grade last year, some had reading level of kinder and some at 6th grade level). Some schools have good community and the kids tend to perform better and their kinders are truly ready for school.
So-
1) check the school and the school district. A lot of district posted their curriculum in the website and see if your DD already knew what kinders needed to know. Then, check if they can do the grade skipping. If the school is awesome school and very competitive, I will let her stay in K. If the school is not that, I would request a grade skip, granted that she completed K curriculum.
2) if she is doing chapter book independently and more than 2 digit addition, subtraction, she will be ready to skip to 1st.
But bear in mind that grade skipping is not always easy. If your DD is mature, she will breeze through it. My DD7 was born in March and she grade skipped to 1st as well as my DD9 (born in October). My DD7 has no probelm and the principals asked me if we wanted to skip one more grade but my DD9 may come home with B grade sometimes (since 3rd grade).
Some district test grade skip before the school year (during registration) and some school does it only after 3-6 weeks. There may be Talented and gifted assessment too. The parents nominate and the teacher sign off on it and they will test the kids. At least, it will give you an idea how your DD is and you may decide later on whether to move up or stay on.
The funny thing is when they test for grade skipping, they will not test her kinder level, they will test 1st or 2nd grade level.
Good luck with the school!
Peter