Honestly OP there are things you need to weigh up here.
The two of you aren't having more kids - you're both happy, together, with what you have. There's no intention of having more.
Is your relationship permanent? - you keep asking what if you split. Is this your intent? Or are you planning on being together forever?
Can you live without sex? - condom failures are a very real fear for her. She's had this happen before and emergency contraception isn't 100%. Neither of you want an unplanned pregnancy, no one wants an abortion. Declining sex with you could very well be how this ends.
Do you feel as though it's time to do your bit? - She's been the one managing birth control and the one who's had the babies. She's put her body through a lot to get your family where you are today. Do you think you can give up your ability to have children to make your relationship work?
Have you talked to her about getting sterilised? - This is an option. It will be brought back to the fact that she has already done so much. But if you both feel so strongly about your positions here, this is a conversation worth having.
I wouldn't be as much of a bitch firm as another poster has been, but as a women with children your Mrs will have a good idea of what contraception is available to her and that really is down to her to discuss with her HCP. But it sounds like she's done with it.