Can she be redeployed to some other position?
Keeping her in her present position is cruel to her and a hardship to your organization.
Either hire someone to carry out her duties, or assign one of the existing employees to her position. If it can at all done, have a healthy employee who can do the job and her swap positions, even if the swapping will result in reduced compensation for her.
My logic is, better take a pay cut than a job loss.
Note: your edit of your post pretty much makes my answer inapplicable in this instance - However, for whatever it's worth,my answer may be applicable to an employer with more resources and more employees than your board.
I wonder if she knows of some volunteer or part-timers who can fill in for her on short notice. The other thing I can think of is, how close are you to a university? May be you can have a pool of students that you can call to fill in for her. Or you could advertise in your state's Department of Labor's bulletin boards for a pool of people who can step in for her. I acknowledge that spending a couple of hours to line someone up from one of the pools for her every time she gets sick - that's not a great productivity idea. If your state's health services safety net is pretty good, you might have to consider biting the bullet and letting her go.