....Sorry, I hardly mentioned his mobility problems....
If the arthritis is painful, it will affect his general condition, stamina -everything. Does he have any medication for this? Good medication might make a great deal of difference. Maybe a consultation with the vet and his bloodwork done would help to get a fuller picture of any issues he has.
Don't push him to do what he doesn't feel like doing. Dogs will often let you know, in their own way, if they want to walk further...want to go home...want to play ball etc...don't want to play. They always give signals of one kind or another. There ARE some slightly lazy dogs but they are few and far between. Usually if a dog doesn't feel like walking and playing it's because it doesn't feel so good.
I guess you are already giving him Glucosamine and Chondroitin supplement? Omega 3 can help a lot too.
For building muscle strength in a gentle way (and also suitable for older dogs) -swimming therapy is wonderful. Easy if you have a pool, and the weather where you are is not cold, and he likes the water. But if he isn't keen, or it's cold, I wouldn't do that.
From what you have described, I don't think this is just loss of bowel control, so much as diarrhea, especially as you have said he has very watery movements. How is he during the day? Does he have any diarrhea then?
A dog with diarrhea can easily have a night-time "accident", especially during a deep sleep when the muscles relax.
It might have some connection to loss of muscle-tone, but I would start by treating this as diarrhea.
There is a preparation which helps firm up bowel movements, and also contains a probiotic. But I wouldn't advise giving it to him yet, in case there is an underlying problem which needs treatment first.
So the best thing is to take him to the vet to be checked out. Take a stool sample with you as it may help diagnosis.