The adapter's photo leaves me a bit puzzling. I don't see anything indicating adjustable output voltage, although the description looks fine. It might just be a stock adapter photo, though...
As for the connector - I'd just solder or use some testing probes clamped to the socket. Ordering at Mouser or other sellers like them will always cost a bit too much for what you use it for. Remember that it's all but certain you can really repair it...
If you happen to have a 3.3v TTL adapter, you could start by only connecting its RX (Kindle's TX, plus GND of course) and see if it is 1.8v tolerant on input - you might get an idea of what's wrong by just reading the serial output logs.
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