.243 will absolutely work fine for deer and has a mild recoil. The problem noone has mentioned yet is that if the stock of the rifle fits you the length of pull will likely be way too long for a 9 yr old girl, unless she is real tall or you are real short. Even the youth model stocks of some actions tend to be a bit too short for grown men.

I would look into getting her a single shot youth H&R in .243 they are very affordable. Then also getting yourself a good used remington for shooting varmints. That way your each have a rifle that fits and will be more accurate shooters for it.

Hope that helps