Changed power socket but nothing works

Not sure if this applies, but if any of the ribbon cables were damaged, or if not fully seated, you will see the DS power up and shut down immediately, or power on with backlight and sound but no images. If you get power and the battery charging light comes on when you plug the unit in, then you probably have this issue.

I've seen a couple of 3DS systems with this issue. My solution was no to fix the socket, but to charge via the "contact points" on the back - rather than using the actual power socket. (look for a dock for your system)

the contact points seem to run to a different part of the system than the power socket.

if that doesn't work either - I'd try to find something similar to nyko's Charge Base for the original 3ds (it has it's own charging system - separate from the 3ds)