She does sound flaky but if you don't want to find a new therapist, you need to take more control in this situation. I don't think it's you and, in the long run, you need to make sure that the relationship between you works for you too.

I have a client that kept forgetting and not showing up. Yes, it put me out that I would show up without fail and it would turn out she was in the car, completely oblivious to the day and time. As a result, I started sending her reminders a couple of days before and asking her to let me know by the end of the next day if she wouldn't be able to make it that week. This has worked much better. It meant that, if she can't make it, at least I know and can plan something else in.

there are two of you in this arrangement and you do deserve the respect of decent communication.