Disclaimer: I personally think marriage is a lifelong commitment, so I'm not gonna suggest you leave her. However, I completely agree with everyone else here that she's being an ass. Money is one of the 3 main reasons why couples devorce (kids, money, sex) so figure this out asap.
If you chose to stay, make a super clear budget and enforce it for BOTH of you. Add up your total expenses per month, tack on an equal amount for spending money for both of you, and have that amount in your bank on the first of the month. Eg. $2500 house, $600 groceries, $800 insurance, $600 gas, $500 spending x 2 people = $5500 a month (all numbers are terrible estimates, don't quote me). Have $5500 in your joint account on the 1st of the month so that you can pay for all your expenses. Take out $500 each in cash, and that's what you spend for the month. Credit cards should only be used on gas, ENFORCE THIS. Once you run out of spending money, you run out. At the end of they month, you should have about $8000 in your account from income. Transfer $2500 to a savings account so you're back to $5500 for the next month and start again. This needs to be enforced for both of you for it to work. Spending amount is negotiable, but it has to be equal. Make the sick cat be a general expense so she can't use it against you for why she has no spending money.
She has no reason to be home if she's not caring for kids. Tell her she needs to get a job, no buts about it. Tell her that you expected your marriage to be a partnership and while she's not expected to make the same amount as you, she's very much expected to pull her weight. Having an additional income could double your spending money if you want, or double your monthly savings.
Lastly, don't begrudge her the money is she starts working. My husband brings in at least $1000 more than me each month and we work equal hours, that's just how it is. We work together on finances and our success benefits each other. You should both be working with the other's well-being in mind, that's part of marriage. Good luck!