Yes, I too had to go and deal with a sick kid and some house stuff. See priorities.
Do NOT tell her you will see about getting the afternoon off, just try. If you do, what a wonderful surprise for the kids.
The email is good.
I like the boundary you set for yourself. You can read them, take the time to decide what to say if anything, and deal with it later.
If she is mad that you didn't reply right away, don't argue, don't defend, simply let her know you were tied up at work and got to it as soon as possible.
Drew--yes Jack's words really are ones to remember and do your best to apply.
"Acceptance doesn't mean resignation. It means understanding that something is what it is and there's got to be a way through it."--Michael J. Fox