like dark said open it up, if you can get or make (never tried making) a triwing screw driver. I've done repairs and soldering on my ds, so i can tell you what to sort of look for. The Shoulder buttons are attached with a pin and a spring, the spring press up on the inside of the button and attaches lightly to the ds on a small plastic cradle. The button, when pushed down, presses on a click button soldered to the board.
Sounds to me like the clicker for your button may be broken. If you can find a site selling a replacement button you could solder it on, if i recall thats not the most difficult component to resolder on.
If your not sure what to look for inside you can also check out a nintendo ds disassembly guide.
WORD OF WARNING! if you take your ds apart be very careful of the two sliders, power, and volume. Tape them with tape to the body of your ds so they dont move, and double, triple, quadruple check they MATCH THE INTERNAL variable resistance slider or you will be sorry. I broke my power switch like this, and replacing the power switch is one of the harder pieces to find and replace
Come to think of it, I havent really seen any in depth take apart guide that shows the shoulder buttons. If you'd like so close up pictures of the button and how it should work let me know.
Edit: Posts merged together. Don't double post. - Ace