How do you get the kinks out of a flat gold necklace?

Honestly, it's really difficult to fix kinks yourself on flat chains (e.g. herringbone style). You could try to force them back into place using some flat-end tweezers or, if you really want it fixed, walk into a jewelry store and ask if they can do anything and what it would cost. I myself find that my old kinked flat chains are best sold for scrap gold and have avoided these styles of chains for this reason.