Though time goes by but many users just like you will meet this kind or problems. In my opinion, a Nintendo DSi just like a storage device if you store pics on it. That means if the storage device lost files can be recovered, so does the Nintendo device.
But I need to remind lots of the users that if data loss happens (no matter happen on the storage device or the computer hard drive), don't put any new files into the device which the lost files were stored before because the data overwritten situation will happen.
What is data overwritten? It is simple. When data loss happens, the files are not really completely deleted from your hard drive or device, only the space of the deleted files are emptied so that there is enough space for putting in the new files. At this moment (before putting new files into the device), the lost files can be still recovered. But if new files have been put into the device, they will take the space of the deleted files, at this moment, the lost files cannot be recovered.
So if you meet data loss problem, the first thing you need to pay attention to is don't put any new files into the device until you find the effective way to recover your files.
As it is a Nintendo device or storage device, I suggest users to plug it in the computer and then find some data recovery programs to scan it for the lost files, once the data recovery software can find out the lost files, there will be a big chance to recover the files.