By the way, I don't think you need to connect your Connection ID. I'm sure doing so may have benefits, but I don't think it matters in this case. I know some people who got the letter without linking their account.
Also, on the topic of receiving gifts, if anyone has the official guide go to page 58. If you haven't noticed yet on the list of possible HRA points, it lists "Digitial Distribution" and describes it as "Furniture that can be downloaded from the Nintendo Wi-Fi connection." We can hope for a lot more stuff in the future.