I'm not sure if the official System Transfer app requires a functional SD card slot. It might be okay, especially if you don't have any DSiware installed.
Otherwise, it might be best to start from scratch. You can manually recover your old save data from your SD card using ninfs on a PC if you have your movable.sed, and then restore it to the new system using Checkpoint after you install your games again.
In theory, it is possible to overwrite the movable.sed on your new system and then keep using your SD card as-is, but there are complications to doing that. I'm a little fuzzy on the details.