Does Nintendo even do actual repairs on warranty claims? My experience is limited but I am under the impression that most claims just get a refurbished system sent back.
I once sent in my GameBoy and they basically sent me the latest hardware GB. But it requires paying the repair/shipping cost. It was better than buying a new one, and it had better hardware than my original!
So yeah, they'll upgrade/replace you if under warranty, but it's cost saving for Nintendo to simply replace it and scrap your old system. So don't expect your 1st born LE whatever to return to you in working condition.
They'll just pull extra stock off the shelf and throw yours on a pile of "rejects".
Man, I'd love to work for Nintendo and actually repair those systems. You know how much fun that would be?
Oh, and they do this for your plush Pokemon as well. What they do not fix is the dog who eats your plushes out of spite. The warranty doesn't cover "the pokemon that ate it". And if it's a LE like a gold Magikarp, go ahead and cry.
WHAT YOU CAN DO:
Go buy a new one. Switch the LE casing. Send in the new one with bad hardware under new warranty and get another system that works! You're still buying a second system, but why not double up instead of down?
Whatever you choose, I'd be pretty worried about sending an Arm9loaderhaxed system back to Nintendo. Don't feed the trolls...
Otherwise, go get a soldering gun and learn to solder. Go buy a replacement port and good luck! Might as well Hard Mod the thing while you're at it.
Or find someone who will do it for you... This is GBATemp. Offer to cover shipping + repair costs and I'm sure someone will do it for you...