How do you switch mii's from one console to another?

My first wii is broken and I'm going to buy a new console. But I can't store the mii's or the mii channel onto the sd memory card, and I hear that by switching mii's with the controller doesn't allow you to edit your mii or have full access to it. How can you switch the mii's in a way that they are truely owned by the new console?