I was actually just having the same problem, which is how I ended up stumbling upon your question. After a lot of frustration, I realized that the two little tabs on the rear of the button were being blocked from sliding all the way into their receptacles because a little bit of plastic had bent to block the opening. Once the openings were cleared the button worked properly again. Hopefully this photo makes things a bit clearer: http://i.imgur.com/SuYXBe5.jpg Just make sure there's nothing blocking the slots circled in red.
I know you posted this a few days ago, so perhaps you've already answered the question yourself, but if you haven't, the problem is the speaker cable. It's pinched, frayed, or may even have a tiny tear in it. You can fix it yourself, but it's certainly one of the more irritating repairs to do, so you may want to have a professional do it. The nice thing about doing it yourself though is that it's actually very cheap. You can buy just the ribbon cable itself and reuse your original speakers to save money, rather than buying the cable with new speakers already soldered on. One bit of advice though is that if you buy a third-party speaker cable from eBay you should buy two just to be safe. Sometimes they're faulty straight from the factory. If you buy an original cable taken from another 3DS console though it should be fine to get just one, since the seller should have tested it. Here's the guide: Nintendo 3DS Speaker Assembly Replacement
If the screws are completely stripped this probably won't help but I used to have the same problem until I started using WD-40 on all the screws in the lower housing before trying to open 3DS consoles up. I spray a bit into a plastic cup, dip the tip of a straw into the cup and drop a tiny bit into each screw hole. After leaving it to sit for about an hour the screws are much easier to remove, though they still require quite a lot of force.