It's absolutely none of her business. I have a policy of not disclosing any non-essential information about myself to any Korean employer, current or otherwise. This applies to all personal information. It is simply none of their business. Her asking you for non-essential information after you have left her employ is especially unacceptable in my opinion. Moreover, she's lying about immigration needing it. Immigration has the information, and any further information they need can be gotten directly from you. The staff at immigration speak English. They will call you. They won't go through a former employer.