I think the BT Sport app is now Chromecast enabled. A Chromecast costs about £30 and plugs into a spare HDMI port on your TV. It connects to your WiFi and you can then display BT sport from the app.
Apple TV can be got for about £64 from amazon. It works really well and is very straightforward to stream from your iphone/ipad to TV. If you set up home sharing on your PC and Apple TV box you can listen to your itunes music/watch purchased movies/TV from PC to TV.
If you have a look on ebay you'll see that the 2nd generation Apple TV boxes sell for much more money than the latest 3rd generation model because the earlier model can be 'jailbroken' (to run a lot more applications than apple supply) but the latest can't. At least not yet.
I know that you could also use chromecast which is much cheaper but it does seem much more temperamental.
If your TV has HDMI then Apple make a Lightning to HDMI wire link for the usual £25. You'd also need an HDMI cable and to adjust the aspect ratio on the TV but it works fine.