I went through a similar time a couple years ago when my wife was hospitalized for weeks on end, then required in-home nursing care for several months following, and I was the primary caregiver doing tube feeding, IV nutrition, medicines every 4 hours, etc. All the while, I was working, trying to hold down the fort with my three year old, trying to get some sleep, and trying to maintain my sanity.
As a caregiver, you will learn sooner or later that healthy self-care is critical to being your best self for your wife and children. Sometimes that means putting something “important” off for another hour of sleep. Sometimes that means outsourcing things that you “should” be able to handle (laundry, cleaning, yard work, etc). Sometimes that means being anywhere but home for a couple hours and giving yourself permission to be “off duty” mentally or physically.
When my motivation waned, I signed up for a fondo six months away so I had some external forces acting on my will to train. I also found incredible benefit in a Friday morning group ride (social pace) that always hit a donut shop. My love for donuts matches my love for bikes and so that one was a great weekly motivator for me. I was able to do that ride because my daughter was in daycare and my wife was ok for a couple hours at home, though I did have some backup help planned for piece of mind (mine and hers) while I was gone.
I also found my neighbors and close family to be helpful. Neighbors asked what they could do to help - I kept a short list I could hand them and say “nothing is expected but anything on this list would be great”. Things like mowing the lawn, cooking a meal, picking up a prescription from the pharmacy, and having my daughter over for a play date.
Sometimes a 5am trainer session is better for you than an extra hour of sleep. Sometimes the opposite is true. Do whatever works for you on the day, with the knowledge that it will likely benefit everyone the next day.
Good luck, and I hope you check back in with a progress report when you can.
Tyler