We have tried to talk to her about this, but she acts like there is no problem. She says she doesn't want to be treated like an old lady, but she seems to have trouble with normal tasks. Her house is quite a mess and nobody wants to visit there. I thought about calling the board of health, but my husband won't allow it (he helps mom with bill paying and other paperwork since she loses and forgets things easily). He is worried about offending her, i think. As you can see, this matter (as well as residual issues) keeps me up at night.
does mom have dementia? Tell us more about her. She probably needs help with many things some of which are obvious, like the dog, but others you haven't even guessed. You should start with taking her to her doctor, and then have a geriatric assessment done. Who has her POA's? If that hasn't been done, I would take care of that immediately.
Dogs can be "litter box" trained. Having a potty station may be a big relief to the dog when no one is available to take him out when Nature calls. If your mother is not able to take care of the dog, it may be best to re-home him with someone more capable.
I wish your mother would take the dog out more. If he was house-broken in the past, it may be all that the dog needs to stop his indoor-pooping ways. Her taking him out would help her with her depression and the dog in being able to stay in his home.