I would add only one piece of advice to the response you've already received.
We can almost always, with certainty, understand the core source of the frustration and come up with a solution.
We cannot always understand with certainty, what will make the angry person feel better...unless we ask.
I do not know what the circumstances were, so what I'm about to say may or may not have helped the situation, ok?
The customer came in, clearly very upset, which probably led to the abusive language. First thing I would have removed her from that precise location...you don't need the disruption in the workplace for everyone to be spectator, and you don't need to expose other customers to this.
So I would have asked her to step into... an office, a conference room....heck, even OUTSIDE or a restroom (if no other privacy available) to discuss the situation.
Once she had a chance to vent to me privately, I would have grabbed the opportunity of "silented vocal chords" to recap what I heard. Ms. Jones, I just want to be certain that I understand exactly what you meant (experienced, whatever). What I heard you say was ________. Is this accurate?"
As soon as I established what the entire problem was, I would have said something like, "I believe that I can resolve that for you by doing __this__. If I do this, will it help?" And this is the whole key of my point. Outline the solution and inquire if it will satisfy the person involved. Solutions not only have to resolve the situation, they also have to appease those who were affected.
Then, because this customer is already on TILT, whether the solution takes 5 minutes, 30 minutes or twenty-two days, I would tell her that exactly, "Ms. Jones, we've already agreed that if I resolve the situation this way, it will satisfy your needs, right? It will take me about 30 minutes to accomplish this. Would you care to wait, or shall I call you when it has been fixed?"
If time and travel are involved, and the customer starts to tilt again at this point, and depending on what resources are immediately available, I would make an offer like this: Tell you what, Ms. Jones, this is going to take me about 30 minutes to resolve. Rather than waiting here, could I buy you lunch (coffee and pie, whatever) in the restaurant next door? I will come over there as soon as this is resolved."
If she is the letter writing type (as apparently, she was), you want the letter coming in that says "I was really angry and upset, but (insert name here) and her team really understood and helped me get this fixed."
I hope this helps.