This is a very clear cut case that a bank would support you with a charge back.
You did not receive the service that you paid for. She broke the service contract. You, I presume, have written communication evidence that she did not provide what you had paid for and that you have approached her and failed to receive a satisfactory outcome. If by any chance you haven't had these conversations over email, do it now. Write in a formal and factual manner to her, stating that she did not complete the agreed work and that you require a refund from her for services NOT rendered. Put in writing to her exactly what you booked, the timeline of events, and the outcome, and that you are not happy woith the outcome. See if you can draw her into saying or agreeing with you that she did not complete the work - maybe ask her first, in writing, why she did not complete the service, did not give the colour enough time to develop and sent you home with wet hair (even though she's told you why in person).
Then you take screenshots of the conversation, the one with the salon, and the one with the payment processor, and send them to your bank when you submit the dispute. Assuming that you do online banking, you find the transaction and there will be an option to dispute this transaction. Or give the bank a ring and do it over the telephone. In my experience when you have evidence like this they put the money back into your account immediately and inform the vendor. It is then up to the vendor to prove that they did no wrong - they have something like 30 days to do this I think, In the event that it goes their way (it won't), the money is removed from your account again. Often it is the bank, rather than the vendor who has paid you back, but they have all sorts of things that they do with regards to the vendor - if they have too many chargebacks, the banks can flag the transaction as suspicious to future customers at the time of booking, for example.
Please do charge it back. £85 is not an insignificant amount and it is SO clear cut that you should not have been charged the full amount. I don't think chargebacks deal with part refunds, and I REALLY wouldn't sweat it, as it's 100% her fault that she didn't refund you 90% or whatever herself - well, the whole thing is entirely her fault, so she can go whistle for that few quid for what work she did do, and for the charge from the payment processor, and for the reputational damage with her payment processor (who aren't going to be happy with her, as the chargeback is damaging to them).
And, as others have said, absolutely do leave a truthful review wherever you can online. Be factual - the facts don't need any embellishment, they're absolutely awful, and she can't get a factual review removed from most platforms.
I would also do the same for the payment processor and for the salon. The payment processor/booking system should have a system for dealing with complaints against hairdressers, they have failed you, and people deserve to be forewarned that any money they pay through them is at risk. The salon, if you booked through them, or if they advertise her as part of their team, are just as guilty in failing you and also deserve to take the reputational damage.