I'm so relieved to read that you are not going to do this. I am amazed and appalled that this lady had the nerve to ask a casual friend to volunteer her time, money and resources to this extent for someone she barely knows. It would be a heck of a lot to even ask a best friend or beloved, close family member - much less a friendly acquaintance. (!)
That said, I'm a little concerned that Mary might be avoiding your phone call, knowing that you're likely calling to back out. I think she knows exactly what she's asking of you, but she's hoping that you're going to be too nice to "let her down." As in - if she avoids you long enough (and avoids getting you in touch with Cici), it will be "too late" for you to decline, and since you're such a nice person, you'll just go ahead and do it for her.
I can see her coming from a mile away. Sure, she may be sick - but she doesn't have to talk - just listen. Heck, I had to go to the emergency room Christmas night with a sinus infection from the depths of Hades, but I still wasn't too sick to pick up the phone to find out the wait time. She can pick up the phone to hear you say "no." She just doesn't want to.
So, I strongly recommend that you avoid the phone call altogether, and email her a polite "I'm sorry, but it just won't be possible for me to coordinate your daughter's wedding. I couldn't get in touch with you by phone, so I wanted to make sure to let you know as soon as possible. I hope your daughter has a beautiful wedding and that you feel better soon. Best wishes, Region."
That way, you also have an email trail that you DID contact her by such-and-such date in case she wants to turn around and blame you for anything. I know it doesn't seem like she would, but trust me, I've seen it happen. The nicest people in the world can surprise you the most with their nastiness...I know from harsh experience that the one person I thought would never turn on me...did.
You can always make a follow-up phone call to ensure she got your message, of course. But in either case, I would strongly recommend that you send her something in writing, either via email or snail mail, just to cover your you-know-what.