First, I'm assuming that the credit cards are personal. I'm also assuming that the business was not incorporated.
Second, is this a separate credit card, not joint? If you are telling me that these cards are joined in some fashion but that she obtained a credit card "without your permission," you may have a problem with liability even though it doesn't show up in your credit report. Your signature may not be required. Additionally, she is the primary debtor. If your cards are linked then don't be surprised if the credit card company comes after you since you would also be jointly liable according to the contract. If you don't pay when you might be legally on the hook, then it is only a matter of time until it is reflected in your credit report.
Third, filing for bankruptcy requires some serious thought. Don't be surprised if your wife's bankruptcy doesn't show up in certain matters concerning you. This is a serious matter and you'd truly need to talk this over with an attorney or bankruptcy specialist for a full consultation. The facts here are of critical importance.