My opinion is to let her tell you when she's ready. If you let her shoot a gun that has more recoil than she's comfortable with, she'll develop a flinch. And they can be hard to get rid of. Also, be sure to teach her all the dynamics of shooting. All children and most adults don't have a clue what happens when a projectile leaves the barrel ofa fire arm, and there is a lot more than most people think. This way if she's missing high or low, she'll be able to explain why, and that she's probably right on the money. It's pretty cool to see children gain confidence in shooting, but that comes with good sound dicision making on your part.
I have a Chinese made SKS that is a great little rifle for kids and adults. It is a 7.62 x 39 cal, which is a great whitetail deer round. Some will say it's not powerful enough, but if you take responsible shots, I assure you it is. In fact since it isn't too powerful, I think it may even be delivering more energy into the animal than my 30.06 which is probably guilty of putting 3000 ft lbs of energy into the hill behind the animal instead of in the animal. I've dropped a number of deer with it, and it works just fine. It's an auto loader, so the recoil is drastically reduced, has some cool history (it is a combat rifle comparable to the AK-47), their pretty inexpensive. I paid $100 for mine and it was new. But that was a number of years ago. I think they go for around $300 now, but they come with a sling, ammo belt, oil can, cleaning kit and tools. The best part is - they are so cheap to shoot. A box of "wolf" hollow points is only $2.10 for twenty rounds. And that is the same ammo you'd use for hunting. And with ammo that cheap, your daughter will be able so shoot that gun enough to get really good with it and it will really boast her confidence.
But let her find her own pace. I have four kids (two boys/two girls). My oldest is nine and he's a boy. I took him hunting when he was seven, and shooting a ground hog ruined him for a couple of years. We were deer hunting, but on a piece of private property, and one of the rules is "ground hogs are to be shot on site". This year he says he wants to go again, but he may still not be ready. But man he can shoot, so when he does decide to go - deer look out!

There is a really good book called "shoot better II". The first chapter tells you basically everything you need to know about shooting, and it is not filled with a bunch of boring filler. The rest of the book is filled with tables on ballistics. So after you decide what you want out of a gun/cartridge you can use those tables as a reference.
Good luck!