Hey mate. To get your DSi repaired, there are 3 options.
Option 1: You can go to this link; http://en-americas-support.nintendo.com/..., which is the American (I'm sorry if you are not from America, you can change the location, if you are not) Nintendo website. You simply;
1. Accept Nintendo's Online Repair Terms and Conditions
2. Log in or set up a new repair profile so we can securely communicate with you about your
3. repair
4. Check the warranty status of your system
5. Confirm the repair, shipping, and any billing info
DONE!
Nintendo will fix your DSi for you, but as this isn't a warranty fix, it will cost you. as for how much, I'm, guessing $65 + (+shipping)
Option 2: Google stuff like Nintendo repair, and find an electronics repairer that will do it for you. Its most likely going to cost around about the same as Nintendo though. You might even have a local repair shop (sometimes computer or mobile phone repair shops will do it).
Option 3: buy a new card reader and replace it. The problem with that option is it will require you to de solderer the old one and re solderer the new one on. Its not normal soldering either, as it is an SMD component, making it hard to do (and I think they use lead free solder, so you would need to obtain some). If you know someone that can do SMD soldering, maybe get them to have a look at it. A new reader would cost maybe $10 + $2 for led free solder.
I hope this helps you out.