Yep, this is the place to vent as we have all been there, or at least ones who have been going long enough. However, you will get the odd ones coming on and saying "suck it up and move on" but I'm with you on this one.
As long as you keep your response professional and polite, then all is good. You've explained how you have covered your basis for her to know those details and if she contacted you via your website, then that's another tick on your side! I would keep all correspondence as proof that she agreed to the clean and then send her an email outlining that you will take this further and use a debt recovery agent which can hinder her credit rating etc etc and incur further costs. To prevent this from happening, please make payment within 14 days.
I had a customer who owed a couple of cleans and I left him text messages and notes through the door. I finally sent him an email as I realised he paid on PayPal once and so I had a way of contacting him further. So I highlighted the cleans and money owing and that I would take the matter further, as mentioned above. He paid me within 4 days and had a good moan, somehow it was my fault that I couldn't wait longer than 2months to be paid 😂😂.
Boohoo.
I got paid, he got upset n threw dummy out and nothing came of it. Apart from he's the only house on the street which I don't clean 😂
In this case however, yes people in the street may or may not know what character she/he is like. They normally get away with it or people can't be bothered to chase up on it and they may be reliant on that, which gives them a certain level of arrogance and for that reason alone, I would make a professional stance and follow through.