I bought a pack of these, and they were incredibly tiny. I just measured then and they are 3mm x 5mm for the plug , and 3mm x 6 mm for the jack.
https://www.amazon.com/gp/product/B01DUC1KEO
I did 3 pin, because I planned on powering the ftx with a battery, not the kindle.
I can measure the jack part when I get home too, to see it it would fit in the open area.
I decided against using them originally, because I was planning to solder them directly on top of the serial pad, facing up. but the amount of force required to unplug the cable was too much for the solder to handle. but if you have the corded part in the case, it may work.
Here is the spec for them:
http://www.jst-mfg.com/product/pdf/eng/eSH.pdf
All of this type should be 3 mm x <something base don number of pins.>
Could you fit a flat flax cable inside, folded up, like these guys?
https://www.amazon.com/Waevshare-Ras.../dp/B01LY2V23F