Support for the Kindle Fire emulator seems to have been discontinued as of sometime in early 2014. You can still find the page describing it here on the Wayback Machine, but now it is gone from the Amazon developer pages:
https://web.archive.org/web/20130919204543/https://developer.amazon.com/sdk/fire/emulator-guide.html
On this forum page:
https://forums.developer.amazon.com/forums/thread.jspa?threadID=479
an Amazon developer rep finally states that "We have dropped supporting Emulator. Sorry for the inconvenience."
The Amazon developer site's "Pre-Submission Assessment" page says to use an actual Kindle Fire device for testing:
"To complete this group of tests, install the app on a Kindle Fire tablet and launch."
https://developer.amazon.com/appsandservices/support/submitting-your-app/tech-docs/03-pre-submission-assessment-guide
They don't actually say "we had an emulator, but it didn't work out," but as we've been left to draw our own conclusions, that might be a pretty good summation.
For its new Fire phone, Amazon has made available a testing service which uses an actual time-shared Fire phone to which your APK gets uploaded. The test routine will push some buttons and make some random pokes at grids and the like, and some random keyboard entries, and then a few minutes to hours later you'll get a link to the results (including actions performed and their relative times of occurrence, and the resulting screen shots, plus a logcat) on your developer account. It works but it is by no means a comprehensive test.
https://developer.amazon.com/public/resources/development-tools/app-testing-service
For my part, I have just purchased a Kindle Fire HD 2013 solely for testing, used, with a tiny crack in the corner of the display but otherwise working fine, for a very low price. It's always better to have a real device than an emulator anyway.
I'll probably supplement that with some regular AVDs that are as close as possible to the various other Kindle Fire devices.