It's horrid when this happens. I've been 12 years in this business and have worked out how to deal with complaints. I used to be a national trainr so have a deeper experience from that too. Here's my advice.
1. This matter is nothing to do with insurance. Insurance wouldn't cover this anyway and there is no legal requirement to have insurance. It is to safeguard you in case an accident occurs that you could be sued for. I'm not saying not to have any, it is sensible to have some, but there is no panic and it would not have helped here, at all.
2. Put some terms and conditions on your website on a page by themselves. You are welcome to check what I have on mine. I don't know if my website appears in my signature or if you might have to message me for it. But definitely include in these: 1. that you cannot deal with any complaint if the fitting has been dealt with by any other fitter. All complaints must be referred to you before they are removed or otherwise interfered with. this makes sense since you can't deal with anything you can't see or that anyone else has had a hand in. 2. That any chemical treatment of the extensions is made at the client's own risk and that you are unable to take any responsibility for the hair condition from that point. (always advise clients that there is a risk to colouring extensions and that your supplier tells you to advise clients not to do it) 3. The law for services is different than it is for products and the law says that you are allowed to make right anything the client is complaining about. If you choose to give a refund instead you can but the client cannot choose. not when it's a service. This means you could change something wrong with a fitting, add more hair if clients says not enough - she'd need to pay for hair - go back to do more blending etc. You do want your client to be happy but you do not need to do whatever an unreasonable client requires. The law protects you from that.
You can see that these three points would disqualify her from a refund right away.
Always behave respectfully and politely in response to a complaint and express your regrets that she is unhappy but be firm about what you are prepared, or even required, to do to try to make things right. When I've tried to fix the situation, if it becomes impossible as it has in this case, then I write what I call to myself a termination email. In it I repeat how sorry I am and list my own case against her complaints. In this example I would include the colour damage, her testimonials, the time delay, remind her of any advice I gave her at the outset, the fact that once the extensions are removed I cannot help, I would point her in the direction of the terms and conditions Then I finish by repeating in a formal manner that due to these facts I will not be offering a refund. I want the client to see what I would be presenting as evidence in any case pursued against me. Usually that puts it in perspective for her and I've never had a client take anything further. I work very hard with any reasonable complaints, of course, to make my client happy. And in fact I try my best with the unreasonable ones but I don't ever give a refund. The work has been done and the time has been spent.
If she has been rude or threatening in any way then I say this...
Since nothing I'm saying to you seems to help you and what you are saying to me I'm finding unpleasant I will be terminating communication from this point on.
then I block emails, texts, calls, facebook and whatsapp. My address will be on my website so any legal communication would find me. I don't see why my heart should be in my mouth every time I look at my phone in case she's continuing to harass me.
I do hope this helps you and I wish you the very very best in your new career. You are doing well to have taken action that can bring you in a good income. continue to learn as much as you can. watch professional videos about fitting, blending, cutting etc and take any good advice people give you.
Bless you.