Okay, I'm not sure if you have fixed it yet, but getting it repaired is pretty easy if you know what you are doing. Trust me, I have helped many people get their DS repaired when something like this happens.
I'm from the US, so I'll be explaining this from that point of view.
-Step 1-
First, you need to know some basics about your 3DS. When did you buy it? What happened to it? How is your system responding after it broke? Those are some things that you should mention when you contact them.
Probably the most important thing you need to know is how long you have had it for. If you have had it for less than a year, then you can skip the next step. If you have had it for more than a year, then follow what you have to do next.
-Step 2-
If you have had it for more than a year, then you should see if you can get your warranty extended. Hopefully, you still qualify for an extension otherwise repairing might get costly. A warranty extension involves a Club Nintendo account. I'll tell you now, if you already have a Club Nintendo account, and you have already registered your 3DS, chances are you don't qualify unless you have registered it within the past three months. You can check the status of your warranty through your Club Nintendo account (if you already have one).
If you don't have a Club Nintendo account, make one. You can start by registering your 3DS here
https://club.nintendo.com/todo-product-register.do
Where it says "PIN #", put the serial number (located on the back of your 3DS) into the blank. It will then ask you to sign up, or sign in for a 90 day extended warranty.
After that's all said and done, you should be set for some more warranty time.
-Step 3-
Now you need to make a repair ticket. Yours will probably most likely be under "3DS Hinge" so here is the starting process for the repair:
https://store.nintendo.com/ng3/repair/repairTerms.jsp?sys=3ds&sym=HINGE
They will ask you to agree to some terms, then they will ask you to make a Nintendo Repair account. It is NOT linked to your Club Nintendo account, so you will have to make another one.
After that, follow everything that they want you to do. Don't hesitate to contact them if you need any more questions asked during the repair process.
I hope this helps!