It's hard to tell from the picture, but does it look like it might be coming from his vent? With that pinkish coloration, I'm wondering if it isn't a prolapse - it happens sometimes. . . basically their intestines. . . come out. One of those things where it may fix itself or not, but as far as I know there isn't very much you can do. If you have aquarium salt it wouldn't help to add some to the tank (assuming that his tankmates wouldn't mind it). If it's squishy looking, that may very well be it. . . hopefully he'll be okay. Is he eating well and behaving normally, or is he seeming very stressed out?
YOU sound stressed out - sorry to hear that the missus is sick. I know how difficult it can be to deal with sick family AND sick fish at the same time - hopefully everyone will be feeling better soon!