The Nintendo DS has two builtin fuse, labeled F1 and F2. These are engineered to help protect the circuitboard. Occasionally they kick into action and what you are left with is a NDSL that charges but wont turn on. Or wont charge or turn on.
Most likely the fuse is gone/blown. Open up your NDSL and check the F1 and F2 fuses with a multimeter for continuity. ie if there is resistance, then fuse is OK. If there is no resistance then fuse is Blown. Occasionally theres protective lacquer layer and you need to stick probes in deep to get true reading.
Cheap work around is to Bridge the fuse, so there is continuity. This is a workaround and onetime fix. Next time your NDSL has a surge, it will be your board that will be blown. Best practise is to replace fuses with fuses.
Do this at your own risk. Tin a thin wire and bridge the fuse on the board.
The proper way to fix it is to replace it with a SMD 0603 Fast Blow fuse. Recommended to use the 1A rating ones.
Its not that hard to remove the SMD fuse if you have a steady hand and paitence. You can use a small flathead screwdriver to help lever it off the board while you de-solder.
It was hard and tricky to get the new fuse in the right place as it was small and hard to align.