Those clear things mounted at the RX and TX labels are LEDs. The one labelled RX turns on when the RXD line is pulled to ground. The problem is that the kindle 1.8v TTL cannot sink enough current to reliably pull that line down to a logic zero. On mine, I removed that LED and replaced it with a 22K resistor, and I put a diode in series between the RXD line and the kindle (with the band painted on the diode on the kindle side). That way the kindle TxD can pull the output to GND, or allow the 22K resistor to pull the adapter side up to 3.3v or 5v, depending on how is is set. See the "super simple" serial level shifter thread. You might not need the extra stuff that connects to the kindle RxD. Be sure to use a known good ground at the kindle, like the shell of the usb connector. The resistor needs to be about 20K or so to work properly, but some variance is allowed.
Here is the level shifter circuit:
And its theory of operation: