As for the recommended amount of sleep, sources vary - most state between seven and nine hours for teens, with a rather new study suggesting that the lower amount may be linked to better academic performance.
If you read them carefully, you will soon notice that individual needs can vary and current recommendations usually add the caveat that each child must be seen as individual. Not unlike for adults: some will be happy and awake after five or six hours, others tend to be in a "zombie trance" if they don't get nine or more hours.
So in theory your daughter might be fine with seven hours, but from what you write, this seems to be not the case:
she always seems to be really tired and that is affecting her mood
Note thas she is at an age where she has a lot of "growing" going on, both physically and mentally. Especially her brain is right in the middle of a major re-structuring process, going from child to young adult. Sufficient rest and above all, sleep, is important. A lot of wild things are happening in her brain while she seemingly "does nothing, just sleeps".
You seem to have a strenuous schedule and I'm not convinced that it is just the lack of sleep that wears her out. We do have a "don't argue with the premises" policy here on Parenting SE, but allow me to play devil's advocate here. Let's just take all "she has to" points:
has to get up at 6:30
Can you shave off just a tiny bit of her morning routine? I know she's fifteen, but how about less / quicker make-up and hairstyle? Can you pack her lunch instead of her doing it herself? Perhaps (and I don't really like the suggestion) breakfast on the train instead of at home?
has to take the train
Ok, probably that is fixed, but how about using that time for homework? Reading exercises, vocabulary and studying for tests all don't require a desk and might be more effective at 5pm instead of at 11pm.
Or this can be a dedicated time to relax and socialize, especially if she rides together with friends.
has dance practise every other day
This makes three afternoons, if I calculate correctly. Would her performance be so much worse if she does only two? You obviously can't build strength without actually practising, but mentally rehearsing or going through the motions is a proven technique to practise without actually doing the moves. Many professional athletes use it and I can confirm that it works. Again: think of those two hours on the train. Note that excessive training may increase the risk of injury, destroying all the work that alrady went into her dancing.
has to do piano practise
Again: can you reduce the frequency? Alternating between dancing and piano sounds reasonable. The point of excessive strain on her fingers is the same as above.
has to do homework
Yes. And no. Did you consider cutting down the effort she puts into school? Sometimes it's a question of efficiency, not of the sheer number of pages written etc. Have you talked to her teachers whether she can reduce the workload if the dancing and piano are a serious career option? You might be able to work out an arrangement or find out that you consider school more important than dancing and piano.
has to do dance and piano as she's auditioning for performing arts summer school
What would happen if she either doesn't audition or fails? How "good" is she already with regard to what is expected? Ask yourself: how important is that summer school for her future life? I don't think she will be a professional piano player and dancer in the future.
Your daughter is fifteen. At that age, you should discuss the points above together as a family and decide which route you want to take. As I see it, carrying on as is will probably cause too much stress on your whole family (if one member is constantly grumpy, for example). If she chooses to keep up her schedule, it will cut into her "relax" time in the evenings because she aparently needs more sleep than she's getting at the moment. Remember that watching TV is not supporting good sleep either, finding other ways of "winding down" would be better.
