These kinds of issues can be sticky. Let's face it, in this country, if someone wants to sue a company they will, regardless of how careful you are about handling their employment . . . but if you follow some basic steps, your company will be protected as much as it can be.
1. Is there written policy regarding absenteeism, or disciplinary actions in general? If there is, you must follow it uniformly to all employees. It doesn't sound like there is any policy written in this case, so then you go to step 2 . . .
2. Look at any similar circumstances in the company's past and how they were handled. If this person is in a "protected category" (meaning basically ANY group other than white, straight, males) then you must be absolutely certain that you are handing out discipline that is consistent with previous practice. For example, if you gave 5 verbal warnings and two written warnings to a male employee in a similar circumstance but then fire the female employee with less verbal or written warnings, your company could be liable for a discrimination claim. Previous circumstances establish a standard practice for handlings that you can be bound to follow for all employees where there is no written policy to follow.
3. If the employee is in a situation that is clearly unique and there is no established standard of disciplinary gradient to rely on, then you can handle her how you see fit . . . just be sure it is a standard practice your company will be willing to continue in the future for other employees, because now you will be establishing the "norm" for how these circumstances are handled. So, you can terminate her if you want, but that means you must be consistent and do that with anyone else in the same boat.
Probably, your best bet is to issue a written warning, with explicit description of what is expected for her going forward. Make the description as tough as necessary. And make it very, very plain: The employee will be replaced if she cannot carry out the functions of her job consistently. And a re-evaluation will occur at a specific date, provided satisfactory attendance is shown. Employment courts generally frown on absenteeism and will back up an employer on these issues as long as they are well documented.
As for the human aspect, I always approach these things with alot of compassion. I have an employee in a very similar situation. We have given her alot of leeway and understanding, but recently, her manager and I sat down and let her know that although we really feel for her, and want to support her in any way possible, we also have a responsibility to our company (as she does) and unfortunately, buisness is buisness, and we NEED her to be here on the job consistently. If she can't, we understand, but the company must go on. For the sake of all its employees and customers, we cannot continue to simply do without her on the job. So as hard as it must be, she needs to find a way to be on the job. And the company is not being unreasonable to demmand that. Especially after all the courtesy she's been shown to this point.
So just be sure that what you do doesn't go against established practices in the past . . . and be very honest and transparent with the employee. Make your expectations known, understood, and written. If she blows up and quits, so be it. You are not being unreasonable to expect that an employee show up for work.
If you do this, the likely outcome is that she'll try hard for a week or two and then slip back into the habit of missing work, and if that happens, you MUST follow through on your warning exactly as you said you would. Then there is no debate.
Hope that helps.