My 3DS got wet, what did I lose and what can I recover?

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.

Disassemble the device and brush the PCB with a soft toothbrush and isopropyl alcohol (90%+), wash the PCB with alcohol again and let it dry for about 12-24 hours (alcohol damps fast).

It might fix your issues. We have a ~66% success rate with this method in our repair shop.

I went through a similar experience a few months ago, and have two things to say, both to the asker and to anyone else seeing this question:

First off, even if it's been a week and your 3DS won't turn on, don't lose hope. I dropped my 3DS in a sink full of water while it was turned on, and although I immediately turned it off, removed the battery/game card/SD card and dried it out thoroughly, It took about ten days before it would turn on again. I recommend leaving the battery out of the system and letting it dry separately, so as to reduce risk of anything weird happening as it dries out. The power button and the touch screen are probably the last two things that will return to working order, which is probably for the best.

Secondly, the other answers and comments about alcohol are absolutely correct- I will elaborate on them slightly here. Once the water has evaporated from the system, any remaining problems are due to material that had been dissolved in the water and now is interfering with the electronics, or with corrosion of the surface of the electronics. Isopropyl alcohol (rubbing alcohol) is a fairly powerful solvent, capable of dissolving most of the contaminants that water can leave, and evaporates quickly and cleanly. If you are willing to partly disassemble your 3DS (no harm will come to the data by doing this, though there is always a risk of damaging the hardware further) then I would focus firstly on cleaning any visible rust or corrosion (I noticed some near the headphone jack and around the game card), and then on the shoulder buttons if they have problem. Screens and buttons aside, the electronically sensitive parts of the 3DS are remarkably well shielded, and there will most likely be no problems with them. If you have a 3DS XL, you can easily and safely disconnect both shoulder buttons and clean them, as they connect to the main body with a simple snap-on connection (I believe the ordinary 3DS only has one shoulder button connected in this way, with the other being more securely attached, but I could be mistaken).