I've been looking at getting myself a pump-action gallery gun in .22 for some time now. I've done lots of looking and fondling and very little buying. All of the .22s I've ever handled have short length of pull, including most of the autoloaders, bolt-actions, pumps, and lever-actions. At 11, she should be big enough to handle a regular .22 without any real trouble, unless she's just a really small girl. Frankly, I'd recommend a cheap single-shot bolt gun like the M44s and Yugo (or Czech, can't remember which country they're from now) .22 trainers. I like the idea of the red dot sight, but I still think it's a better idea to make her use iron sights to learn the basics. Get a cheap scope and tell her that she can have it if she learns the basics, kinda like a reward for doing well. Don't make her wait long before giving her the scope though, no more than 2 or 3 range trips.
After she has learned the basics move her up to a lever or pump gun. Just from personal experience I believe that it's a bad idea to give an autoloader to somebody who hasn't learned absolute fire discipline. The lever or pump has the added advantage that she'll be able to shoot .22 Long and Short out of it as well.