In certain school districts if your child will be 5 before December 31 in the year you wish them to start kindergarten they will test the child to see if they believe they are ready for kindergarten. I know our school district (Valley View Community District 365U) has this policy. I don't know how difficult it is to have your child start, but I would think that if they completed a program in a public school setting they would be allowed to continue to the next grade. I am not certain, but if they do end up allowing her to begin the program, she should advance yearly if she meets the criteria.
On a side note... if you are going through a separation you may want to think twice about pushing her ahead. The separation could be hard on her and the kindergarten program might be too much for her to handle along with the separation.
Here is what the Illinois School Board of Eduacation has to say on the matter:
(105 ILCS 5/10‑20.12) (from Ch. 122, par. 10‑20.12)
Sec. 10‑20.12. School year ‑ School age. To establish and keep in operation in each year during a school term of at least the minimum length required by Section 10‑19, a sufficient number of free schools for the accommodation of all persons in the district who are 5 years of age or older but under 21 years of age, and to secure for all such persons the right and opportunity to an equal education in such schools; provided that children who will attain the age of 5 years on or before September 1 of the year of the 1990‑1991 school term and each school term thereafter may attend school upon the commencement of such term. Based upon an assessment of a child's readiness to attend school, a school district may permit a child to attend school prior to the dates contained in this Section. In any school district operating on a full year school basis children who will attain age 5 within 30 days after the commencement of a term may attend school upon the commencement of such term. The school district may, by resolution of its board, allow for a full year school plan.
(Source: P.A. 87‑359.)