This app was mentioned in 6 comments, with an average of 1.17 upvotes
Old tv- irrelevant. Firestick has the tech you need built in.
Newer firestick: this should work: https://www.techfollows.com/firestick/how-to-cast-to-firestick-using-android/
Older model firestick or dongle: https://play.google.com/store/apps/details?id=de.twokit.video.tv.cast.browser.firetv
I happen to have the somewhat older 4k capable dongle and use it on a regular 1080P tv. It works for this. Not perfectly 100% of the time from all sites (sometimes the video format is incompatible), but still very good with full definition and no picture artifacts (as you would get in, say screen "mirroring"). And good enough for me not to take a new firestick (4k ones have better hardware/RAM) out of the box.
Lets you easily use a very wide array of anime websites with shows (say you want uncensored Interspecies Reviewers and your fully kosher Funimation and VRV subscriptions don't carry it, or you want an OVA that you can't get, or even something on YouTube to go to your tv).
https://play.google.com/store/apps/details?id=de.twokit.video.tv.cast.browser.firetv There's one for app
This app can cast any web video. works really well. Future reference now obviously...hope u figured it out
This one works for iPhones https://itunes.apple.com/us/app/iwebtv-cast-web-videos-to-tv/id999462129?mt=8
Looks like your main question was answered but I'd also like to say that I've had good luck with the "Video & TV Cast for FireTV". I haven't used it in a while but it works pretty well. It can cast most videos from their built in browser. I used to use it quite a bit before youtube had native casting.
I haven't tried these methods, but from what I've seen you should be able to pull this off through either the Kodi machine or the Fire TV. There are apps/addons for both which claim to allow you to stream any video (same as a Chromecast would).
https://www.amazon.com/Video-TV-Cast-Fire-web-video/dp/B00V4V3TWA
https://play.google.com/store/apps/details?id=de.twokit.video.tv.cast.browser.firetv&hl=en
https://www.amazon.com/ClockworkMod-AllCast-for-Fire-TV/dp/B00JHMPP9S
https://www.howtogeek.com/277116/how-to-cast-youtube-and-other-web-videos-to-kodi-like-the-chromecast/
1) Re. NHL video -- have you tried Video and TV Cast for Fire TV? It's from a family of apps that install on Android phones and send an artificial cast/fling request to streaming devices like the Fire TV for the streaming device to begin pulling in video content.
2) Re. your music collection. Have you tried the Amazon Music app? I believe the app should find the music files on your phone SD card and I think it will be able to "fling" (Amazon's version of cast) the music to your Fire TV
3) Re. podcasts. Most podcasts are actually just MP3s. So after your podcast app pulls them in, your Amazon Music app should be able to find them, and 'fling' them as mentioned above.
4) Re. general mirroring to the Fire TV. Have you tried going into the Fire TV settings for Settings > Display & Sounds > Enable Display Mirroring? Then if your phone supports full screen mirroring, you should see an option I think in your settings or pull-down nightshade settings. Mirroring is slower and shakier than casting/flinging but you won't be limited by the content source
5) Have you considered a Chromecast also? Unfortunately a lot of apps only built in casting for teh Chromecast and never added fling capabilities.
Malware scans for the linked app(s):
de.twokit.video.tv.cast.browser.firetv: ApkScan | VirusTotal
de.twokit.video.tv.cast.browser.lg: ApkScan | VirusTotal
de.twokit.video.tv.cast.browser.samsung: ApkScan | VirusTotal
de.twokit.video.tv.cast.browser.roku: ApkScan | VirusTotal
de.twokit.castbrowsernexusplayer: ApkScan | VirusTotal
de.twokit.castbrowser: ApkScan | VirusTotal
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