Yikes. This is like looking in a mirror. I actually did something similar about 20 years ago, shortly after I graduated college. I lent my brother $5,000 on my credit card, for a business venture he was attempting, with the understanding that he would pay me back every month until it was paid off.
Well, my loan to my brother became an unmitigated disaster, and while he did eventually pay me back, it took about 12 years. I am really sorry to tell you this, but your options range from bad to terrible. I'm not a lawyer, and so to verify that your options are as wretched as I fear, I dropped a line to the very affable Theodore W. Connolly, a Boston bankruptcy attorney and the co-author of "The Road Out of Debt" (Wiley, 2010). And unfortunately, he agreed with me. There is very little you can do.
It sounds like your agreement was informal and that your brother never officially became an authorized user on the account. Let's also assume you've already taken possession of the card from him, and preferably closed the account, so that no more charges can be made.
Should you take him to court? I didn't think you could sue your brother, since this is your credit card the debt is on, but Connolly actually says you can. In court, you could argue that you had an informal contract with your sibling with the agreement that he would pay off the debt on your card. Then if you won (a big if), Connolly says that you could attach your brother's new house or bank account as a way to collect. But, "It would be an expensive, ugly lawsuit and would really make Thanksgiving dinner uncomfortable," Connolly says, adding that he wouldn't recommend going that route.
Neither would I. You lent your credit card to your brother because presumably you wanted to help him. Trying to make him pay for his mistake may be justice, but it will almost certainly ruin your relationship, and that animosity could spill over to the rest of your family if they start taking sides.
If I were you, I'd keep the lines of communication open with your brother and let him know you're happy with any monthly contribution to the credit card that he can make. And I would consider other sources of income to help pay down the debt, such as taking a second job for a while or seeing if anyone else in your family would want to pitch in and help.
Finally, if he can't or won't pay you and you see no other options, I'd consider talking to an accredited, reputable credit counseling service listed with the National Foundation for Credit Counseling and see if they can help you work with your credit card issuer to lower the payments. Perhaps, as Connolly suggests, they'll be able remove any missed payments off your credit history.
But, just so you know, once you begin officially working with a credit counseling nonprofit, where they contact credit cards on your behalf and negotiate interest rates, that will seriously hurt your credit score (it's not really fair, especially in your case, but that's just how it is) However this works out, good luck.