I'm so very sorry you're both going through such difficult times.
I like the idea that you can do anything if it only lasts for 15 minutes. Aim to spend 15 minutes a day doing something to make it better. If you do more, great. If not that's ok, do another 15 minutes tomorrow.
I'd start with the things that will make the biggest difference to how it looks. Just spend 15 minutes walking from room to room with a bin bag. Throw away all the rubbish and anything that's pointless clutter.
Don't think about recycling, selling or charity shopping anything. You don't have the headspace. Just bin and get it out of your house today
Then walk around again tomorrow with a box or basket and pick up everything that needs putting away. Don't put it away yet. Just put it in a box of doom!
The next thing I'd do is wash any dirty pots, clean the sink and wipe the kitchen sides.
Then I'd put everything away from the box of doom that has a home, and leave anything that doesn't in the box for now
After that I'd sit down with a cuppa and make a list of jobs that need doing every week. I wouldn't bother with any deepcleaning jobs, if you're at the firefightering point. I'm just taking about clean bath/sink/toilet, dust, hoover, mop, wipe kitchen sides, change bedding etc. Then divide them up over the week. Set a timer and spend 15 min a day doing something from the list
Once you've been doing that for a while I'd make a list of the deepcleaning jobs and do 1 a day on top of what you've been doing.