The Fit is a good car for a teen, reliable and efficient. I would get something a little bigger though. IIHS only recommends midsize sedans and larger for teen drivers. We got my 18 year old a 2015 Mazda3 last month coming out of a 2005 Prius. It is much more substantial than a Fit. It is still not a midsize sedan, but it isn't a subcompact car either.
Just make sure your daugter gets all the safety features you guys are willing to pay for. The mandatory feature is Electronic Stability Control, which was mandated for all cars since 2012. A backup camera is also a nice feature. Automatic braking would be out of your budget.
The Toyota Prius is probably the best car for a new driver that doesn't have a lot of gas money. 2010 and newer gets at least 45 MPG; the 2012 and newer gets at least 50 MPG. There is a lot of cargo space and the back seat is spacious. The Prius is super reliable, with most old ones going past 300k miles after replacing the battery once. It is a more premium car compared to the Corolla, Civic, Cruze, and other compact sedans. Active safety features were available since 2010.
My second suggestion is the 2014+ Toyota Corolla. It is a reliable compact sedan that is easy to drive, pretty stylish, comfortable, efficient and roomy.
The Mazda3 is the most stylish option. It is the most fun to drive out of these cars. You could afford a 2012-2013, right before the current generation. Road noise was pretty loud in these and the ride was sort of firmer than the Corolla and definitely the Prius. The Mazda3 was reliable. 2012 and newer had a much more efficient engine.
All these cars are easy to see out of.
Some cars to avoid are the Ford Focus (transmission failures), Dodge Dart (junk), VW Jetta (expensive to fix), 2012 Honda Civic (reliable, but didn't drive well at all), Chevy Cruze before 2016, and any subcompact other than the Fit.