Hi Dan. Just a few weeks ago, I had perhaps the same issue as you are having. There was no sound in IE.
What I needed to do in order to fix it was log into my web host and use cPanel to add MIME Types.
Here is what I had to add and also what my .htaccess file looks like:
AddType application/manifest+json .manifest
AddType application/font-woff .woff
AddType application/x-web-app-manifest+json .webapp
AddType audio/mp4 .m4a
AddType audio/mp4 m4a
AddType text/cache-manifest .appcache
AddType text/xml .xml
Just to be clear. I didn't just add the above lines to my .htacess file but used the cPanel.
Your webhosting provider should be able to quickly help you get going and if not, just log into cPanel and use "MIME" in the search field which should bring you right to the page where you can add these lines ... one by one.
I hope that helps.
P.S. Can I ask you for a quick favour? I don't have access to a Kindle device and wonder if you would ming visiting my website http://littlecomposers.com and click on apps and just try and open one in the Kindle? IF something behaves weird, please let me know. No hurry ... just if have some time down the road. THANKS