I've faced the same question as "Wrestling" a few times in my teaching career. My response in each case depends on the attitude of the student, the particular course structure, how far into the semester the issue comes up, whether I've had enough coffee, etc.
While I completely agree with the assessment of what the student needs to be doing, I would also play Devil's advocate a bit: maybe, even if the student is ultimately going to fail the class--or, as likely, come for a couple weeks and then drift away again--maybe she needs someplace to be for a few hours a day that's not at her mother's bedside.
In my (large) classes it wouldn't be particularly disruptive for a student to show up mid-semester. The grading policies preclude makeup exams, but this is one of the situations for which we could justify dropping the grade on a missed exam, or even two. Realistically, a student walking into my course at this point in the semester is going to stand almost no chance of earning a passing grade, because of the cumulative nature of the material, and yadda yadda; I would make sure to say that to the student up front, and I would encourage her to consider withdrawing for the semester, even talking to her advisor and seeing if there's some way she can get some money back, and re-enrolling for a later term; but I would also say, if she wants to try, she can. This student may have been putting off going to classes one day at a time, and may have just figured out that she's running out of days--things can catch up with you that way when you're in the midst of a personal crisis, especially if you're also dealing with depression--and she may need a couple of days or weeks to adjust to the idea that it's too late for this term. It's hard for a person to just suddenly turn her self-image around 180 degrees; to her it will feel like "Yesterday I was a college student, today I'm not," even though anyone looking from outside could see that really she hasn't been a college student for weeks.
While it's frustrating on my side to spend a lot of mental energy and time on a student who is very likely not going to complete my course, it may pay off in the future, in having her feel it's worthwhile coming back later, when she no longer has the obligation that is currently consuming her life. I wouldn't want her to shut the door on her future education because I turned her away when she was vulnerable.