I understand your desire to pay this bill on your own, but realistically, finding a good work-from-home job is very difficult, and really how much work can you get done with 4 children in the house? Your options are pretty limited, like providing child care or taking in work for a special skill (sewing, computer work, repair services, etc...)
Another route is taking a job that provides free or reduced-cost child care, such as working in a gym, community college, school, government office, etc as someone else mentioned. Or find a job where you can bring your child along, such as driving a bus/van, a paper route, local deliveries, etc... Or can you trade-off child care with another stay-at-home mom? Then you can have regular time available for a part time job while providing the same benefit to another mom.
Finally, and I really hesitate to add this because it might sound judgemental, which is not at all what I mean, and obviously I'm a complete outsider who knows nothing about your relationship or overall situation. But from a total third-party perspective, I always cringe when I hear a stay-at-home parent say the biggest obstacle to them achieving what they want (in your case, an independent income) is their spouse's job. It keeps the SAHP from acheiving some self-sufficiency and traps the working parent in a job that, quite frankly, sounds really unpleasant. Regularly working until 10, having no flexibility in his schedule, even occassional 20 hour days is really hard for a man with 4 young children at home. It seems really unfair to both of you. You're probably both exhausted and that makes it really difficult to have a rational perspective. And it doesn't necessarily get easier when they're all in school. There will lways be sick days, school vacation days, parent-teacher meetings, etc.. I suggest, as part of your "recovery" from this one incident, try and come up with a more equitable arrangement - so that you can earn some income, and so that he can spend more time with his family. If you do seek counseling, this should be part of the discussion. Again, this is just a suggestion and totally not a judgement, just an opinion from an outside perspective.