Are you sure that model is AirPlay-capable? In the web specifications page for that model, the closest thing I see is Google Chromecast, which is similar in concept but not interoperable.
If yours is in fact AirPlay-capable, you may have to enable that function through the menu system on the TV.
We have a Sony Bravia TV of similar vintage which can handle either AirPlay or Chromecast, but it is a different model. It got AirPlay2 support in a recent update.
Some iOS/iPadOS apps can handle ChromeCast instead of AirPlay, while some offer both. Apple tends to focus on their own AirPlay protocol, so builtin screen mirroring and the TV app offer only AirPlay icons. Regrettably, I have no experience with separate apps for enabling Chromecast-based mirroring.