Hm... If Patchy's saying "Obasan" would it be a better translation to say "old lady" or something like that? Well, its hard to find a equivalent.

I chose ma'am because it's respectful (like obasan, and it fits with Sanae asking for a favor) but still indicates that Sanae has called Kochiya old. I don't think "old lady" works as well, but I admit that there's probably better choices.

Actually, isn't it Patchy saying, "Please, Ma'am?" I think it's Patchy asking to stay.

I'm pretty sure Patchy is the one saying that at the end, or else the globe would be pointing to Sanae. And she wouldn't have that face.

So it is more like a little kid calling a woman old unintentionally.

I think you're right. That was supposed to be little Patchouli asking, since Sanae prompts her to in the previous panel. Translation changed.

I think a better phrasing here would be "auntie". It's respectful, but has the implication of age as well, which "ma'am" doesn't really.

Changing it, but feel free to revert it if you disagree.

Auntie seems the most accurate.
Plus "old lady" is "obaasan" with the elongated "ah."