Hey buddy,
Got your text and read what's going on here, so figured I would respond here so I could better articulate a response rather than just some short message....
Though I've never had to deal with a family member having serious health issues and can't even begin to imagine how you're feeling, I can understand the sheer frustration of feeling helpless in a situation and spreading yourself thin trying to be there for your family and maintain your other obligations.
For the sake of being 100% open and honest, about 5 years ago I was going through a pretty horrendous time. Divorce, no place to live, completely ignoring my own self and just spending as much time at work as possible and if I wasn't working I was either drinking or binging on crap food (when I could eat that was). This went on for awhile and I was starting to go down a pretty dark path. One day out of the blue, someone I hadn't talked to in years texted me and all it said was this; "Step 1: Get back in the gym. Step 2: Start eating real food again." Naturally my next question was "what's step 3?" To which the reply was a simple "get through steps 1 and 2 first."
The next day I got back in the gym and it was rough, but I went 5 days a week and got back into a routine. Soon after that my appetite came back and I lost pretty much all desire to eat junk food, and also didn't go days at a time without eating. It wasn't too long after that other things started coming back around; got a promotion at work, started being social again, talked to my friends, started doing things that I normally would never do like yoga classes which is how I ended up meeting my wife, started competing again... The list kind of goes on.
Moral of my story being, when my life was falling apart and I felt like I had no control, lifting weights and controlling the foods that went into my body gave me a center, a place where I could always revert to where I was the only one in control. There were days where time wouldn't allow for workouts, and sometimes I had to break my diet a bit, but that was less than 10% of the time.
So my advice would be, keep the mindset that you're doing the show, eat as well as you can which may take a little extra planning (always carry one more meal than you think you'll need), get to the gym when you can and even when you don't feel like it, because we all know sometimes those days are just pure therapy, and keep taking care of your family.
Hope this helps my friend!