Yes, I have faced this problem many times.
By all means sympathise and commiserate, but be firm: nails have to go.
I am a firm believer that if you have a well defined goal (in this case you have: get the nails short), and if you keep changing your behaviour, it is just a matter of time before you come across a behaviour that will elicit the results you want. The only problem is: do you have a large enough repertory of behaviours? The general philosophy is: if what you are doing is not working, do something else.
By all means, always be patient, but be ready to
pretend to be impatient . You should have no limitations as far as your external behaviour is concerned, but on the inside you should always be calm and collected, otherwise you will not be able to determine the best course of action. I always try to approach difficult cases with detachment and great curiosity: What can I come up with that will give me the desired results? Negative emotions are bad advisers.
Here are some suggestions that you may try: (And remember that what works in one case may not work in another and vice-versa).
1. Buy a set of false nails (really long ones), glue them to your own fingers and play like that for your student. You don’t need to say anything. She will get the message.
2. Encourage her to paint her nails and decorate them in any way she wants as long as they are short (sometimes offering an acceptable option to both of you may work).
3. Explain to her that nails are – biologically speaking – excreta just like faeces and urine. She wouldn’t go around with faeces hanging form her back, likewise she should clip her nails. Also explain to her the hygienic side: how much dirt and germs collect under long nails (The idea is to try to make long nails undesirable – while they are seen as desirable, she will not clip them).
4. Get a book of bizarre photographs. Usually there are some photos showing people with the longest beards, the longest nails and so on. The one I have show this woman who never clipped the nails in her left hand. (She needs the use of her right hand). Her nails are weird callous excretions that twirl around in strange spirals. Discuss with her the pros and cons of such behaviour (pros: celebrity; inclusion in the Guinness book of records – cons: she needs to be extra careful not to break the nails, her whole life revolves around taking care of her nails, she can’t go out, her left hand pretty much lost the movement because the muscles atrophied).
5. Find out what is it that makes her long nails so desirable. Show her alternative ways to obtain the same results.
6. Buy a set of fake nails for her this time. Tell her she can have them and use them if she clips her real nails and don’t use the fake ones for practising/playing.
7. Since she is leaving anyway let her do it her way. But don’t bother to give any instruction either. Make sure you explain it clearly to her: You have been given instructions. I am now waiting for you to follow the instructions. I cannot proceed unless you help me by following the instructions. Give her the scuba-diving analogy: You go to a scuba diving lesson. The boat is ready for you, the apparatus is ready for you, and the instructor is waiting. You were given the instruction to bring your swimming suit. If you turn up without a swimming suit, you cannot go scuba diving. So get a swimming suit. We are waiting.
8. If you don’t need the money, send her her way. Make sure you explain clearly to her and to the parents why you are letting go of her: If she is not prepared to follow instructions she is not ready for piano tuition.
9. State clearly in your policy (I do) that long nails are a definite no no. It may be to late for this one, but it may avoid the problem at the root with your next students.
10. Yes, why not, if everything else fails, clip her nails by force. Here in the UK, however this may result in court proceedings he he

.
These are just a few possibilities out of the top of my head. You may be able to come up with different ones as you talk to her.
Best wishes,
Bernhard.