Am I overestimating how much you're working? Or is the income being consumed by something?
If you can convert three hours of "sitting and stressing" time to two hours of "self and family-care" and one hour of "productive work", then that is an increase of productivity and pay.
Also, as others have mentioned, reading and learning a system is billable time. Programming is thinking, not pushing buttons. (Staring off into space or stressing or distracting yourself with social media is not, however).
It would probably do you well to find a way to clearly separate work and life time/space. Work in a library or a coffee shop away from distractions, and set aside time during the day to be focusing on work and work alone. Take responsibility for controlling what you spend your time doing.
I think Patrick McKenzie recommends contractors bill by the day instead of the hour, partly to get them to consider necessary break time as billable. Of course, if you get to the end of a day and realize you really have been very distracted all day, you don't have to bill it as a full day.
Family distractions can be a big problem, though. You have to have blocks of several hours — I think the ideal might be two 3-hour blocks in a day — where you know you're not going to be interrupted, short of a true screaming emergency. Getting this across even to one's spouse, never mind a young child, can be very difficult. Getting out of the house might be the only way. As Paul Graham once observed [0], even the prospect of an interruption can be enough to keep one from getting into a flow state.
I think your problem is that you don't get shit done because you get interrupted so often and than know exactly that you aren't making any progess and go straight into panic mode.
Start with step 1: close the door and start working.
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I've raised special needs kids. I'm seriously medically handicapped myself.
Unless your children are all under the age of two, there are things you can do to empower them to take care of themselves.
Store things they need where they can reach them on their own.
If you have a microwave, set up microwave foods and microwave safe dishes near it. Let kids feed themselves when they want to. If most of the food at home is adequately nutritious and they are given some simple guidelines, this won't undermine their health.
For example: My son liked eating cold noodles from the fridge with Parmesan cheese. I told him if he had an apple to go with it, that was a perfectly acceptable light meal. He did that on his own often in elementary school.
Give them very simple rules and standards to follow, such as "I need to be able to walk to your bed, dresser and closet without hurting myself." Then let them figure out how to meet that standard.
I homeschooled while bedridden. The kids piled into bed with me and I read to them.
The point of that: Handicapped people (such as your wife) can often do a lot more than other people like to believe they can. Their physical limitations need to be accommodated, but it doesn't mean they can't do anything.
However, they frequently do not have the energy to deal with their physical limitations and also try to prove to other people that they aren't useless. So the attitude and mindset of folks around them matters a great deal in their ability to both contribute and to lead full lives.
I'm the primary breadwinner for my family unit. I work part-time and intermittently as I am able. Over time, I have gradually increased my earning capacity, in part by focusing on getting myself healthier.
In my household, we talk a lot about "primary breadwinner privilege." When I had a corporate job, my adult sons took over the women's work at home. I'm firmly convinced that a lot of men get as much done as they do because social expectation bends out of their way to accommodate The Job, not their gender per se. That line of thinking has been very helpful to me and my sons for trying to find a path forward under extremely difficult circumstances.
So it isn't being included to man bash. It's being included in hopes that it's somehow useful to you as a mental model.