Speak with the Kinder teacher. It's very hard to get 2nd year funing for Kinder these days (where I live, might be different where you live) so you might find that holding her back will mean she's home for a year. Schools probably would take her mid year but then she'd be 6 mths behind all the other kids which could be hard for her.

If she's academically ready then I'd be inclined to send her to school, the social skills will come and if need be she can repeat prep.