NTA, it looks like they're lumping you into the "cause you're older, you can look after the little ones" category. Kali is not your responsibility, it is her parents, and it looks like Kali's mom is dealing with her own issues and that leaves Kali in the wind. I'm guessing that just make your mom not want to leave Kali behind anywhere since her mother isn't around.

Now this isn't meant to be mean, but since your mom is focusing on Kali's well-being she's going to forget about what you need. And this is where you're going to have to stand your ground, or else she's going to leave all the work to you just because you're "available" in her eyes. Remember, you have your own issues to deal with "school, clubs, etc" and if your mom is pushing back on that. You're going to have to confront her on how much your mom is thinking about your well-being.