I think $200 might be difficult for a good bike. I've bought a couple of nice, older mountain bikes on Ebay. After much looking, I ended up spending about $250 for one, and $300 for the other. I did a bit of repair on both and they're now excellent-quality bikes that should last for many years.
I've said this before and I'll say it again: Make sure your daughter is 100% behind any bike-buying decision. If she doesn't love it she probably won't ride it, at least if she's anything like my daughter.
It sounds like your daughter already has the riding bug, which is good. If she likes mountain style bikes, look for a used bike on Ebay. I recommend buying a complete bike, rather than a frame. I've bought 3 frames and built them up. I've ended with bikes I really like, but I spent considerably more than when I bought a complete bike.
Of the two complete bikes I bought off Ebay, one was an old rigid Rockhopper that needed pedals and new brakes. I also put a rack and bar-ends on it. When I finished I had spent about $330. The other was a newer Rockhopper with suspension forks. It needed a new Gripshift. When I finished with it I sold it to a girl on the block for $230. I don't know how well the suspension forks worked, but they moved when she bounced them, so she was happy. I doubt if she'll ever ride it anywhere except around the neighborhood.
My 19-year-old daughter wanted a bike to ride to school when she went away to college. She rejected a mountain bike or a road bike - she'd been there and done both when growing up and had developed a distaste for both. We ended up getting a Trek "comfort bike" with a step-through frame, suspension forks (only good for cushioning bumps in the pavement), a suspension seatpost, and a huge, cushiony saddle. She put a big basket on the front for her books and she loves it! She thinks its "really cute", which is probably at the top of her list of priorities.
I approved it because it has a lightweight aluminum frame and 21 speeds. With her loathing of hills, I'm glad she didn't get a heavy, single-speed beach cruiser.
She rides it so I'm happy.
Good luck, and I hope your daughter continues to ride with you as she gets older.