How can one relate to someone who makes no sense at all? The only thing my husband responds to is, "You are a good man; everybody likes you."

Dementia is a cruel thief. Your frustration is part of your grief.
Good suggestions already posted.
If his hearing is ok, music sometimes reaches through the dementia damage and elicits a response - movement of the head or fingers, small smile, etc.
It is hard to start to build a life without him, but if you can add one or two weekly activities for yourself with other people around, that will help you have people around you, even if not for the wonderful conversations that you once shared with your spouse. Exercise, book group, etc. Also a support group for family members who are caregivers - and you are still his caregiver.

In the later stages of AD, your husband will respond to your body language and facial expressions only; not the words you use which he can likely no longer comprehend. Send him signals through your tone of voice and relaxed attitude, as well as the smile on your face. You are not trying to 'relate' to him anymore; you are just trying to keep him comfortable and happy, in reality. He may not even realize you're his wife at this juncture, but just someone who makes him feel good, which is your whole goal. When he sees you, you want him to brighten up and feel happy, that's all. Positive body language, a sense of calmness & perhaps a hug will do a whole lot more for your husband than any words can possibly accomplish. Put aside the sense that you have to make him understand you, or that you have to have a 'conversation' with him; that's no longer feasible, unfortunately.

Wishing you the best of luck with a difficult situation.

The responses that suggest visiting at mealtimes and taking him out for walks I am going to try and implement. The latter, I think, will be much easier in a wheelchair. Sadly, he used to enjoy seeing our dog, but on the last visit had no reaction at all. I didn't mention it, but my three surviving children are little consolation or solace. Yes, loss of his companionship and fine mind (full-tuition scholarship alum of Harvard College) is damned rough.

I went to my mother’s Memory Care facility today. She doesn’t make or respond to conversation any longer.

So, I just sit in her EZ chair, and read. She mostly stares at the ceiling.

She does thank me for coming, so I know she values the visit. I bet your husband does, as well.

I think you relate in much the same way you do with very small children who haven't yet learned to talk; with touch and actions and you doing all the talking.
I found it easier to visit with my mom during her meal times, I could always prattle on about the food and the people in the dining room and since she needed to be fed it also gave me the assurance she was actually getting one meal a day. Going outside for walks (I walked, she was in a wheelchair) was also a good way to do something for her that she wouldn't have been able to do otherwise, and it was easier to carry on a one way conversation about what I saw.

Sadly, that is exactly what Alz patients do. He isn't trying to irritate you, he simply can no longer follow a conversation.

You treat him like you always did---hopefully that was by being nice. Sounds like you no longer live with him, so you are dealing with 2 losses--his companionship and his mind. No one will say this is easy or fun. You are grieving for the man he was and never will be again. This is very hard.

Trying to have a meaningful conversation with somebody in the last stages of Alzheimer's is a waste of time.

His brain is broken. Do you think he's doing it on purpose?

Try to talk to him as though he is understanding. You don't know what's getting through, so just keep treating him like a human being. Just don't expect responses.