My son is the resident expert - here's his suggestion:
Re-download the games/apps off of the Nintendo Eshop app. Once you buy a game from there, it should be free to re-download it, and most of the apps are free to begin with. The only problem is that it's tied to that particular DSi. If you can't get it back, the games would have to be paid for again. The shop app icon is a blue bag; afaik it can't be deleted.
Regarding the music, to my knowledge you have to add the music files back to it by plugging the SD card into your computer and adding them to the correct folder. Unfortunately, I personally haven't done anything with music myself, so I can't tell you which folder to put them in.