At 230k it owes you nothing... did its job, Keep looking forward :-)
Maybe get her engaged in this scoring/rebuilding process too. Less talk about the negs and more about the things you both (go team) can do together to recover together.
Maybe I am old fasioned, or just different. I believe this falls under the better or worse part of the vows. You could "let her sink" on her own but, ultimately, what does that say about your marriage?
Also, triplets and an extended stay in NICU - has she seen a therapist? There could be other underlying issues there.
I'm not trying to be an a-hole but maybe a couple of things to think about.
Just my opinion....
Credit is earned. At some point, a lot of us were given a tiny little credit line, that first chance, to see how we would handle things. It sounds like both you and your wife were given a second chance with credit. It's up to each of you what you want to do with that second chance.
You've said you are being responsible.
You've said your wife is NOT being responsible. With you making her minimum payments, she is not learning or earning more credit. Please stop paying HER bill for her. She can do it herself.
Not sure what electronics you have at home. I put things on my calendar and set up SOME alerts, and the calendar automatically populates on my husband's phone. The ALERTS will notify him of an event. I put reminders on there, for birthdays, date nights, etc. Sometimes he will call me and ask, "Hey I got an alert on my phoen for x, y, z. I'm on it." or "Thanks for the reminder!" or "Oh yeah, we got that thing tomorrow."
Can you do something like that for your wife? An alert or put the payment due date on her phone/calendar?
We also have a calendar on the fridge. I use it for the menu only. DH often wanders over to read it.
When we had a lot of debt (lots of student loans, mortgage, credit card debt, kids tuition, etc) I made a chart and had it on the fridge. The chart included payment due dates. I would cross it off when I made a pymt. Updated the chart every two weeks or so. You could make one of those for your car pymt and her cc. So what if someone else sees it? So what if the kids ask mommy why she has cc debt or if she's made her pymt yet.