I have no idea how to repair this without opening the console.
When I first read your post I said to myself:
“No, I will not open a perfectly working N3DS.” After a few hours curiosity got the better of me (once again). I opened it to compare it to your pictures and came to the above conclusion: I don't see a way of repairing it without opening.
The hardest part is actually removing that tight small screws without striping them. I was soooo careful and used a perfectly fitting screwdriver
and still managed to f… up one of them badly (now to search for a replacement – this will
suck).
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Thankfully I got it out.
Other than that this repair is pretty simple once you got all screws out.
Maybe you can superglue something in place to hold the button without interfering it’s function… or get a replacement shell (part). No need for full disassembly (only replace the back part).
Things you need:
- good Phillips screwdriver (size 00 I think)
- If this is a non-XL model: triwing/tripoint screwdriver (for removing micro-SD reader)
- replacement back part of the shell (or skill and some creativity to repair the damaged part)
Important:
- After removing the screws: Do try to simply remove the back part. Some ribbons are still connected to the motherboard and have to be disconnected carefully before separating.
- Ifixit.com has some good pictures (just go for the mainboard replacement guide – first part is getting the back side off)
- Reconnecting the ribbons is a little frustrating and may require patience.
If you do not want to do it yourself: Sending it in to Nintendo is a bad idea with a modded console. Maybe some independent electronics/phone repair shop will do it for you if you provide the replacement shell part.