As a fellow parent of keen young musicians, I admire your commitment to supporting your children in music by providing quality instruments. Kudos to you!
I agree with your general approach about providing them with a quality instrument to play on. I think one of the key points people are making is that you don't necessarily need to spend $5K at this point to get your daughter a guitar that is at that high quality. As Souldier wrote above, you can likely find a high quality used luthier made instrument for about half of your $5K. That way if your daughter finds that it is not the type of guitar she wants (different woods, neck shape, etc) you can buy and try something else and sell the first one (or keep it as a back-up). Surely in the NJ and NYC area there are lots of used instruments available. I know that I could find something suitable in my areas for that price. In fact, you might even want to look in Canada since your US dollar buys you a lot more here given the excellent exchange rate.
Just for a data point, here's my story. My eldest son, as a young teen, gravitated towards the double bass. We had rented, borrowed, and then bought a cheaper instrument for the first while, but when it was obvious that the instrument was holding him back, with much advice from his teachers, we managed to find him a good quality bass that was not much more than your $5K budget. Now, 5-6 years later, he's a budding young professional bassist and on full scholarship in music performance at the best university in the county. He's won numerous awards, garnered quite a healthy amount of scholarship $, and recorded a CD. All on what he calls his "first and only bass". We weren't thinking of any of that when we got him the nice instrument - just that he would enjoy playing it more because it had better sound and playability. So who knows what role us getting him a good instrument played in his path so far, but I have no regrets.
All the best in making your decision.
2014 Douglass Scott Cdr; 2010 Martin Blackwell Double top; 1989 Daryl Perry Sp; 1989 James Frieson Sp