Really? I'm surprised to hear that. Are you using a case with the Kindle that is possibly preventing the headphone jack from being inserted deep enough to make a connection?
Is your Fire in a case?
In theory it's a standard jack, but there seem to be lots of complaints about the plug not inserting properly if it's in the Amazon case. You might want to try using an extension lead with a small plug casing.
If normal headphones (e.g. for an MP3 player) don't fit, there's either a problem with the Kindle or - as has been suggested - with the case that it's in.