On your Firestick, go to Settings
Select My Fire TV
Select Developer options
Turn on Apps from Unknown Sources
Go back to the home screen
Select Search
Search for Downloader
Install the Downloader app
Open Downloader
Allow the required permissions
In the URL field, enter the official Kodi download link for Android
Download the Kodi APK file
When the download finishes, select Install
Wait for Kodi to install
Select Done or Open
If you selected Done, return to the home screen
Open Your Apps & Channels
Launch Kodi
If needed, allow Kodi permissions on first launch