Assuming the 3DS cannot be repaired, there's good news and not-as-good news (and maybe a little bad news) I'll start with ---
The good news:
Game Cartridges: all your save data is stored on the cartridge, with a new 3DS system you can pick back up wherever you left off in your savegames. The only thing lost may be streetpass data, and certain things like your friends lists may be cleared. Some games may require system updates or patches from the EShop before they can played.
Pokebank: Once you are able to use your old Nintendo Network ID, your stored pokemon will be waiting in the Pokebank right where you left them, as if nothing happened. Unfortunately, because this is tied to your NNID, this leads us to our not-as-good news.
The not-as-good news:
NNID: Nintendo, unfortunately, will not let you use the same NNID on multiple 3DS systems (I'm guessing to prevent people from buying a game on the EShop once, then sharing it across multiple systems), and since your old (presumably) broken 3DS is still tied to your NNID, you will not immediately be able to use your old NNID on your new 3DS system.
Normally you'd be able to get around this by performing a System Transfer, but if your old system is unable to perform a system transfer due to being lost/stolen/broken there is something you can do:
However, if you give a call to Nintendo's 1-800-255-3700 (6 am ~ 7 pm
Pacific, everyday) Customer Service number, one of the service
representatives can remotely unlink your NNID and make it so you can
re-download your eShop content onto the replacement 3DS.
I have found another article from someone who has gone through the process, and the process will be a bit faster if you have the following information on hand when you call Nintendo's Customer Service:
- The serial numbers of your (stolen/lost/)broken 3DS and your new replacement 3DS. Possible locations of the serial numbers:
- should be on a sticker on the back of the 3DS systems
- on the original box and user manual, if you still have them
- if you registered your system online with Club Nintendo, you should be able to find it online. (Not entirely sure how well this works since they've re-jiggered the Club Nintendo system)
- Your name
- Date of birth
- Nintendo Network ID name
- Club Nintendo ID
- Email address linked to either accounts
- Downloads and purchases you’ve made with the account
- Possibly other information to help them verify that you are the owner of the NNID.
However, once you've jumped through all the hoops (and it can take up to 24 hours for the process of unlinking your NNID from your old system on Nintendo's end) you should be able to use your NNID on your replacement 3DS. For simplicity, I'd advise against setting up a new NNID while you are waiting for Customer Service to unlink your ID, as you cannot have two NNIDs on the same system.
EShop: Once you've gone through the above process to retrieve your NNID, all your digital purchases associated with the old 3DS should be re-downloadable! This should include any 3DS ambassador program free titles, digital games bundled with the old system, and purchases through the Club Nintendo rewards.
The (possibly?) bad news
Unfortunately, the above process to retrieve your NNID is not as thorough as a system transfer would have been. So much of the data on the SD card is probably still tied to your old system. This may mean that you'd have to re-download all your digital purchases. And the save games for those digital downloads? The article I linked earlier suggests that as long as you have your old SD card, and it works, you should be able to use the "Data Management" menu in your 3DS's setting to retrieve it, but I can't find much info on how that works.