Ok, so I am going to slightly revise my advice to begin with reading the “how to convert your SO” sticky. Because given the circumstances you described, you are going to need your wife 100% on board. You work long hours and clearly get home too late to cook dinner, so that is going to fall on your wife. Your wife, OTOH, clearly has the bandwidth to do the cooking: yes, you have two kids, but you are spending $400/mo on preschool, which means she does not have them 24/7 - and even if she did, they nap. And even if they don’t nap, there are literally billions of people who manage to cook dinner with small children around. But you and your wife have somehow created a pattern where she doesn’t do that, for whatever reason.
The thing is, staying home with kids is a privilege. Yes, many families do so successfully, because having one partner take care of the home can free up the other to work harder, get more promotions, etc. But to do that successfully, the SAH partner also has to see her role as a job - a job that includes not just raising the kids, but managing the entire home efficiently. Basically, both partners have to pull their weight financially, especially when the family is in financial straits - both partners need to either add income or decrease expenses. So if your wife isn’t going to add income, she is going to need to work to decrease expenses.
The problem is that going home and telling your wife to start pulling her weight doesn’t tend to go well. 😉. As you will see from the thread I referenced, the only way to make change is to lead by example. So you need to figure out some way to lead your family toward better choices, even though the bulk of the daily work will likely fall on her. So brainstorm some ideas. Maybe raise eating out as a health issue, backed up by a doctor visit as necessary? Or why don’t you sit down with her on Saturday, together come up with a menu of easy weekly meals, and then do the grocery shopping and prep the food over the weekend so she can get dinner on quickly when you get home? Or, if she’d prefer, you take the kids so she can do the shopping in peace (I know what I’d have preferred when my kids were small!). Or, if your wife is financially inclined, you can always sit down with her with the numbers and talk about how much money you are blowing in various categories - then brainstorm the areas you can cut back where the spending isn’t bringing any value.
The other thing I would suggest is lowering your standards, a lot. Two small kids is tough, period. When my kids were small, there were days that “dinner” involved some plain pasta, some cheese, baby carrots, and strawberries. Yes, for the adults too. 😉. Because some days you just can’t, you know? But everyone got fed, no one died, and no one was warped for life because mom set the bar really really low. When you are as far in the hole as you are, you need to look at everything - nothing is sacred. But you also need to have each other’s backs and be kind to each other; it’s a long process to change habits and dig out, but it’s an even longer life, and you want to enjoy that together.