check to see if there's an update for your audible manager. If so, update it.
Restart your computer. Go to audible.com and sign in to your library. Re-download the file you are looking for, then click the download link. It should automatically start your audible download manager.
If it doesn't, go to your downloads folder, and right click the download link admhelper.adh and select audible download manager. This should force the download to start.
If you are only using your audibles on your Kindle, I agree the easiest way is to link your amazon and kindle accounts.