Topic: Is it possible to transfer save data from a broken unit?

I have a New 3DS that it was dropped and the top screen is cracked with the bottom screen also not turning on. The system will turn on but I get no sound or visuals so doing a system transfer through a store seems impossible. I was told I could have my downloaded games transferred through Nintendo support but they told me there would be no way to keep my save data that way. They also suggested that I sent it in to be fixed and I may be able to salvage my saves that way but that it was unlikely. I have everything saved to an SD card but I was told that the saves are partially backed to the system itself. Obviously I'm pretty lost here.