I recently ran into this but the fix is simple and doesn't necessarily require you to take the handheld apart.

3 things to try to get your L and R responding again.

1. blow into and around the shoulder area space. It's as simple as clearing possible dust that might just simply be preventing your l and r button to click down.

2. press the specific L or R button that isn't responding in a rapid motion and try pressing in different spots, like the middle or from the left and right side. (press about 30 times) Try to avoid press it hard where you might risk breaking it.

3. sometimes you can also just kinda press down and listen into to how it clicks and if you feel around while pressing down a few times you can sometimes kinda press it back into sync so it responds again. It's hard to explain this part it but I think you'll get the idea maybe.

I recently just got a DS lite just this past April it was used but in good condition the battery on it still functions. When I ran into this issue I raged untill I found out some ways to "calibrate" the shoulder buttons back to responding to presses as a player would expect.

Had this issue with my DSi's R button. Blowing into it worked for me, for a little while.

All these fixes seem pretty temporary, wish I knew a surefire way to get in there and solder or replace something to get some peace of mind.

Don't blow into it. That introduces moisture, and moisture can worsen microcorrosion.

The best way to fix it is to take the back off, pull the plastic buttons out, and spray contact cleaner directly into the tiny seam surrounding the switch. Then, rapidly press the switches repeatedly.

I've fixed a few dozen DS L/R buttons like this. Unlike blowing into it, contact cleaner keeps it working for a long time.