The app is not available for my Kindle, but I can access Sounds via the website. However, the sound is weaker than on the iPlayer, and I can't actually hear it properly without plugging in an extra speaker. Why do they call it "Sounds" when sound is precisely what's lacking? Fortunately I still have the radio iPlayer, which is fine, and will continue to use that as long as it works.
Another option is available for those who have Alexa on an Echo Dot or similar: it will play programmes from iPlayer if you know exactly what you want and can get it it understand you. I just say, "Alexa, ask BBC to play (eg) The Archers" and off we go!