Jailbreaking a Fire TV Stick is really easy, and you don't even need any special tools for it. First, head over to Settings > My Fire TV > Developer Options, and then enable Apps from Unknown Sources (also ADB Debugging if the option is there). After that, you can get the Downloader app from the Amazon Store.
With Downloader, you can add apps such as Kodi or any other third-party apps of your choice. That's pretty much it; you have now jailbroken your Fire TV Stick, which basically means it can install apps other than the ones from the Amazon Appstore.