You don't need to "soften the blow" OP.
You never agreed to it in the first place, you're not letting her down.
her childcare problem isn't your problem.
do you often feel the need to take on other people's problems as your own? I don't mean that harshly, but maybe this is a really good opportunity to practice setting sane boundaries.
practice saying it beforehand in the mirror if you think she'll ask face to face (blocking numbers etc on phone is easy, it's face to face confrontation that most people will find harder). keep a phrase in your head: "i'm sorry you have this problem, good luck fixing it", or "that doesn't work for me" (whatever you think feels strong/repeatable in your mind).
then just keep repeating it if asked.
e.g.
"can you pick them up tonight?"
- "that doesn't work for me. good luck getting your problem fixed."
"I have a dentist appointment that I can't miss so I can't pick up little Emily, can you get her for me?"
- "that doesn't work for me."
"why not?"
"but you did it before when I asked, what's changed? why are you being like this?"
- "it doesn't work for me."
seriously, eventually it'll go away. (it being the cheeky fucker)