I'm going to throw Rockingham into the ring too as an alternative circuit - its a cracking track and a bit less daunting than Donington for a first timer, although VERY technical with plenty to get your teeth stuck into! Also the banking through turn 1 is a unique experience (and fast). I absolutely love it as a track (its "my local"), plus its usually quite a bit cheaper than Donington for a track day so might give you some leeway for more expensive or inclusive instruction within your budget
Regards coaching, I've found the guys at Racewell motorsport to be superb although disclaimer, I've only used them once for a free 20min session earlier in the year. That said, they're a great mix of Engineering, driving, racing and general hands-on-ness and as an (ex) mechanical engineer myself I found their approach to discussing things like car balance, settings and handling really, really good! They understand it from behind the steering wheel as well as in front of the black board if that makes sense and so given her training your daughter might enjoy that insight too. I believe one of them is actually a lecturer at a college or uni down south somewhere - they're based near Southampton but they travel all over.
They use e36 BMW's primarily, which aren't as quick as the Caterham's suggested above, BUT if this is her first time on track she may appreciate being in a tin box rather than a canoe! I know my wife loved driving a cayman and an M3 on track but hated being in the Caterham, Atom, and Formula Ford on an experience day she did 3yr ago - she spent too much time worrying about being exposed to M3's and things on track around her to really focus and get stuck in to the driving. Its also worth noting a well set-up e36 M3 (or even a 325 or 328) is going to be WAY more capable than she is if this is her first time on track. No offence intended by that statement, but they are very quick cars with the right set-up and modifications and the guys at Racewell spend most of their weekends "taking names" (to use their phrase) mixing it on track with considerably more exotic and supposedly quicker cars!
Racewell Motorsport are on facebook, instagram, etc. - if you speak with them feel free to let them know I sent you. Neither of us will get a discount on anything but its an ice breaker! I have no real affiliation with them, but we have spent quite a bit of time helping each other out via instagram messenger with issues as we're both building and working on the same cars.
Another person to consider is Geoff Steel Racing who's in Kirton Lindsay but again travels all over Europe. Old-school kind of a guy, but a really good instructor with a LOT of experience. Not cheap, but very much worth it and a proper pro-race-team! Obviously something like their E46 M3 GTR is going to be a huge step-up even from a Caterham r500, but they do have other cars they use as well so don't be intimidated by their website! I have no idea what their prices are like, but worth a punt in case they're at a track anyway and you can share with someone else (ie 1 day car hire, 1/2 day instruction). This is actually quite a good way to do it, as being blunt, you're not going to spend the entire 8-10hrs behind the wheel anyway
They are at Donington a lot so very familiar with the territory - my first time on track proper was being driven by Geoff, a wet day in 2001, in someone else's brand new e39 M5, sliding about in the back with his daughter while he tried to figure out how to turn off the traction control!
I have no real affiliation with Geoff - I know his youngest daughter, and brought some parts from them over the years, but never really had anything else to do with them
One last recommendation (or pair of) - Jade Edwards or Abbie Eaton. Jade had a proper filling-rattler of a crash racing in clio cup at the weekend so might be out of action for the very immediate future - they were on ITV's "DRIVE" show a year or two ago as instructors; Jade teaching Johny Vegas and Abbie teaching Prof Green (who won I think). They're both AWESOME drivers with lots of experience, and having a younger female instructor might be nice for your daughter? Both local-ish to Donington and also travel a lot.
I'm not sure either of them have cars they use for instruction, but I'm sure they have cars available via other companies such as BaT, etc. as they will be well known in those circles. I have no affiliation with either of them, just followed them on social media since trying to convince my company (to no avail) to sponsor Jade a few years back
Hope that helps, and let us all know how she gets on!