YANBU.
I'm sorry for her DC being ill, but the issues in her private life are not your responsibility, and her problems are not yours to solve.
Don't be emotionally blackmailed, say you genuinely cannot do it.
I understand what you mean about struggling to get everyone together to meet. My (4) close friends and I work different days and hours, and we don't all live near each other - the closest one to me is 1.5 miles, but the farthest one away is 25 miles away.
And in the past DH has worked odd shifts, DC has lived and worked 25 to 30 miles away, and they (and their partners,) have weekend hobbies etc etc, and it was a struggle in the past to meet up with them sometimes, and we sometimes only saw them twice a month.., and they only lived 20 to 25 miles away!!! (Both live 8-10 miles away now, so it's better.) 
@jillandhersrpite
I think you have to say straightaway - "I know I am always the one to be flexible but on this occasion I have a commitment which has taken a long time to arrange and cannot be cancelled so unfortunately I can't do it this time. Sorry but please remember I have already swapped for you xxx times in the last xxx, so its not because I don't want to help, its because on this occasion I can't."
This. Say this. ^ Brilliant.