I just emailed them requesting that they move my NIID from 1 3DS to another. originally I asked whether it would be appropriate to send the broken 3DS alongside the working one but they just did it digitally.
All I know is that you'll have to pay for the repair cost if you don't have warranty
The reason why I didn't buy a new 3ds and stuck with my old old one was because I know the new 3DS is coming out soon.
Honestly, if I were you I'd just wait until the new 3DS comes out. Does your broken 3DS turn on? Would you be able to perform a system transfer etc?
well I snapped my 3ds xl in half and I contacted them and they were able to erase my NNID off the 3DS XL so that I was able to sync it with a different 3DS and reobtain all of my games
Honestly, I'd just send it to Nintendo. Do you still have warranty on it?