Does her teacher have guitars handy in a variety of sizes for her to try? If not, get thee to a shop so your daughter can a few on for size.
My wife is 5 ft 7 in tall, and finds a full-sized dreadnought guitar uncomfortable to play, preferring a 1/2 scale to 3/4 scale guitar. Her main instrument these days is a Cordoba Mini II Bass, which she took home after getting the chance to try it out at Guitar Center. It's a half-scale guitar, and comes in classical nylon 6-string as well - we're considering picking one up to replace the 3/4 scale Bridgecraft guitar I literally picked up for free at the side of the road a couple years ago. Cordobas are pretty reasonably priced, my wife loves her bass, and I've got a similarly-sized Cordoba baritone ukulele I absolutely love.
With that said...I'd avoid steel-string acoustics for her first guitar. I don't play every day, and there have been long stretches where I don't play at all. Same's true for my wife. As a result, the absolutely lovely Washburn acoustic she has, and my Squier acoustic both tend to sit unused, because they quickly start to hurt our fingers to play because we don't tend to build up the calluses we need on our fingers for them to not hurt. That's not been an issue with her bass, my ukes or cheap-ass nylon-string guitars...
...or, for that matter, my wife's short scale Squier Mini Strat, which is basically a 3/4 length Stratacaster, and was also picked out by my wife after finding it more comfortable to play than a full-scale one. So, starting with an electric guitar *is* possible, if you start out with strings that are aimed at being more comfortable for the beginner. Plenty of people start with electrics, and that might still be a better way for her to go if that's still her end-goal.
Back to classical guitars for the moment, which is basically what you're looking for if you're looking at nylon-string guitars, another brand to consider is Yamaha. They're popular with music schools, IIRC, make tons of instruments in your price range at a variety of sizes, including 1/2 scale guitars, or even smaller - I've got a Yamaha GL1 guitalele, which is basically a 1/4 scale guitar.