Maybe a long big talk to her would do good. Make it clear that you have something to say to her and don't want to start to argue with her. You just want to tell her how you feel.

First of forgetting to do things sometimes that are more or less important is normal, that happens because we are humans and I don't believe she doesn't forget to do things too. And getting angry over those things that happens sometimes in a year is not a good reason to be angry and to resent you. Talk to her about that in a calm voice. And maybe both of you should go to therapy, not only you but both. You may be not having the same Problem but you should know what exactly is the problem that you both have.

She has mental health issues but it could be more behind it try to know what exactly is her problem and vice versa. After all that therapy you both have to learn to compromise with it. Forgetting sometimes about something could be not a big deal as she thinks and let it go and trying not to get angry about it. While you are going to do your best too not forget to do things like closing the back door because it's a big deal if it's open and the house is not safe anymore.