My biggest issue with your T. is that she tells you so much about why she's cancelling and the issues with her daughter. Therefore, you feel you need to put her first because of what her daughter is going through. (because you're thoughtful, caring, etc.)

If she never told you about the difficulties and had to cancel, you would be feeling different. Yes, you may still be hurt that she had to cancel but it wouldn't have the added layer of being about her daughter. If she had told you that she's cancelling to go somewhere, you would have a completely different reaction.

I think it would be better if she stopped talking about her daughter and gave her clients plenty of notice to cancel. So, a little bit of planning on her part and less disclosure.

Hang in there! It's nice that you can still email her. If you keep asking about her daughter (being the sweet person you are), you will probably keep hearing about her. What if you stop asking? As hard as these little ruptures can be, they teach us a lot about ourselves, our relationships and things we need to discuss in therapy.