You need to understand that with no means of supporting herself financially, once she crosses your threshold, any time limit you place on her staying with you goes out the window. I speak from bitter experience! You can say "a month" but once that month is up what then? You ask her to leave, she says she has no money and nowhere to go. What then? Are you realistically going to put all her stuff in bin bags, chuck it out on the street and change the locks? Sure, that's what posters on here might tell you to do but few would in reality. As difficult as it might be to tell her you've changed your mind, it will be a lot more difficult to tell her to leave your home once she's got her feet under the table and insists she has no other options.
Let's look at the facts: she has fallen out with her family, and has been fired from her job recently. Could be she's just down on her luck, but the fact that you spent one evening with her and found her company to be emotionally draining probably tells you something. This is not someone who has her life together. The odds of her finding a new job, saving enough money to afford her own place and securing a tenancy in 4 weeks are extremely low.
The fact that she is unemployed is a very good reason to back out of this, because it changes the arrangement from what you agreed to.
Message her now and get it over with:
"Hi X, it was lovely to catch up. Best of luck with the job hunting. I'm sure something will turn up.
Unfortunately I can't offer you a place to stay while you're looking for work, as we don't know how long it will take you to find a job and a place to live, and I'm sure you'll understand I've got a lot on my plate with the kids and everything. If you get any interviews lined up and need a place to crash the night before you'd be very welcome."
If she takes offence then so be it, it's not like you were that close before so you haven't really lost anything. Besides, the chances of you maintaining a positive relationship once she's installed in your house, living rent free and using you as a housekeeper/Mum/unpaid therapist are pretty slim anyway!