I started kayaking flat water in an old Perception Dancer when I was 9, and learned how to roll when I was 11 or 12. I learned how to hand roll when I was 13 or so. I continued practicing my roll and my hand roll for years at summer camp (two weeks every summer), until the end of last summer when I went out on Chili Bar for my first white water run.
From personal experience, I am very glad I was able to "perfect" (500+ alternating rolls consecutively) a flat water roll before going into whitewater, because when I did flip I was very calm and rolling was second nature.
As for teaching your kid, again I learned at 9 and I subsequently taught kayaking at the camp when I was 16, 17 and 18 years old. The absolute most important thing is for them to have good, positive experiences with kayaking (whitewater or flat water). Campers who freaked out repeatedly while doing wet exits ended up dropping kayaking from their list of sports they wanted to learn. As a young kid, it is scary being upside down in cold water, so all I can say is progressing slowly and steadily will help your kid learn the best and have confidence. On a side note, your daughter will love the sport more if it is her passion. Especially with a sport like kayaking, it is important for her to want to get on the water, not just following you.
I've started teaching kids how to roll as early as 10. I've had campers learn it over the course of two weeks, and some campers never understanding the body positioning or staying calm after working on it for three two week sessions each summer. Everyone is different.
I'd say that flat water is the best place to start. If your daughter knows that you paddle whitewater, let her ask questions before you start bringing up the idea that she should come on a trip with you. Having the personal passion develop will help ensure that she develops her own opinion of the sport. It's invaluable that you are being cautious about not throwing her into whitewater, because even one bad experience early on can cause a younger kid to write off the sport for life.
If you'd like to know the name of the camp, feel free to shoot me an email, firstname.lastname@example.org
This camp is the reason I love white water kayaking so much now (although only having been in it for less than a year; I would have loved to get on whitewater earlier, but I didn't know anybody in the sport and the cost of gear was too expensive for me to front from 13-17 years old.) If you want your daughter to love the outdoors and at the same time learn the basics of how to kayak (from wet exiting to hand rolls/freestyle), I'd love to talk to you more about the camp.