I don't have a Toshiba Fire TV, I have a Fire TV Stick 4K, but does your remote have a Fire TV home button that you can press and hold?
For me, that brings up an option for Display Mirroring, though it's not compatible with any devices I happen to own.
I ended up installing AirReceiver, which supports Google Cast: https://www.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS
While there are several AirPlay apps, which is what you need, they pretty much all have stability issues because they're all just hacking Apple's closed/proprietary protocol. That said, the one I use most is called AirReceiver. It's still very flakey, but at least it has been around since 2014 and the developer keeps updating it. I've found it works best if you manually launch the app first before you try mirroring to the Fire TV.
If it's a YouTube video, your Fire TV Stick should show up to be able to cast it from Chrome to the YouTube app on your Fire TV Stick. To generally cast the contents of tabs (or your entire desktop) using Chrome, you need to install an app. The one I use is AirReceiver: https://smile.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS/
Showing a non-Pi solution here so bare with me.
There is an Android App called AirReceiver available for the Amazon Fire TV stick
https://www.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS/
I run it on multiple Amazon Fire TV Sticks around the house and it allows for flawless AirPlay of Video content from any iOS device to these sticks. It even autostarts and is always available in the background.
The same app also makes the stick a renote playback device for YouTube content, a uPnP target, DLNA target and other things.
We use it to browse through photo galleries, housing ads, car listings, furniture catalogs, hotel/Airbnb listings etc while allowing the family to watch along and decide/discuss on the options shown on screen.
Using it as an extended monitor for a MacBook via AirPlay also works like a charm.
There is also AirReciever. At least the version from the Amazon Appstore has an option for Google Cast, which at my last test worked on the FireTV. Unfortunately this does not seem to work on the Capsule.
if you have a Firestick just buy this app for $3 directly off the amazon store it works fucking nice. I'm mirroring my iPhone 6 ios 10.2.
Ok, after doing a google search, I found this link for AirReceiver https://www.amazon.com/x9F99-x65B9-x513F-AirReceiver/dp/B00L5HQRGS Note that this is not perfect and doesn't support everything an apple tv supports. If wifi speed is not very good, mirroring doesn't go very smoothly. DRM content such as amazon video or netflix won't work, but these are already supported by fire tv natively. I use it mostly for videos from Safari, or just music playback.
It's not the Chromecast but if you have a Firestick just buy this app for $3 directly off the amazon store it works fucking nice. app link! I'm mirroring my iPhone 6 ios 9.3.3 currently and it works without an issue.
Look for a free app called AirReceiver in the Amazon App Store.
https://www.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS
ok now on a pc and looking at the web version of the stores.
Google PLay: https://play.google.com/store/apps/details?id=com.softmedia.receiver
Updated: July 18, 2020
​
What's new in this version
Now to wait when kid is off the TV so i can purchase an test
It's worked great for me for years.
Get this app from Amazon store on your new fire stick. https://www.amazon.com/龙方儿-AirReceiver/dp/B00L5HQRGS/ And it will give you proper screen mirroring for iOS and Android
AirReceiver is another one that has worked for me to cast in various ways, including casting the entire desktop: https://www.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS/
I think it’s this one https://www.amazon.com/%E9%BE%99%E6%96%B9%E5%84%BF-AirReceiver/dp/B00L5HQRGS
https://www.amazon.ca/dp/B00L5HQRGS/ref=mas_ya_dp
I use this app to mirror my phone to my fire stick it works pretty amazingly.
There are other ways to mirror that aren't hardware dependent (edit except for the processing power) You could use an airplay app and buy a license for airparrot windows. You could also try youmap chromecast receiver and mirror the screen using the Googlecast in chrome.
Both apps have free trials for you to test them. http://forum.xda-developers.com/hardware-hacking/chromecast/app-youmap-chromecast-receiver-android-t3161851
http://www.airsquirrels.com/airparrot/
Another is Airreceiver, which recently released a companion app for windows called Airsender. Probably the least expensive alternative option.
http://www.amazon.com/x9F99-x65B9-x513F-AirReceiver/dp/B00L5HQRGS
Dev's website with AirSender App http://www.remotetogo.com/home/?page_id=27
I've bought and tested most of of the AirPlay apps. My top pick is AirReceiver. It's been the most stable doesn't lag (when mirroring) as much as other apps. 2nd place goes to AirPlay/DLNA Receiver because I like the notifications that it provides.
For casting to my fire devices , I use THIS. It's $3 but worth it.