I'm a big fan of Fuji bikes when you're looking for something this small. They run from about $500 up to $1,300. The Japanese know a thing or two about building small bikes...think about it.
The only hard thing is finding a Fuji dealer--that can be challenge if you don't live in a large city.
My 9 year-old daughter rides a 44cm Fuji Roubaix Pro with 650c wheels and it fits her perfectly (once I traded out the 90mm stem for a 60mm). Her brother rides mountain bike and she wanted something different to do with Dad. We ride every morning before school/work on our trainers in the basement for 30 minutes (usually watching one of her cartoons) and every weekend on the local roads for a couple of hours.
As for the rides themselves, you've got to make it fun. Don't push her too much...only a very little to slowly gain strength. Any more specific questions I'm happy to answer.