I realize the Tiled API is not completely intuitive. That's something I've been working on improving for the next version.
One thing that might help is if you have a read of the TMX Map Format page. The C# classes follow a very similar structure. In other words, if you open up your TMX file in a text editor you might be able to figure out where your animated tile is located in the code.
For example, the
TiledMap class has a
Tilesets property that is a collection of tilesets in the map. The tilesets are named based on the texture they use. So you might be able to do something like this (off the top of my head):
var tileset = tiledMap.Tilesets.FirstOrDefault(t => t.Name == "mytileset");
var myTile = tileset.Tiles.FirstOrDefault(t => t.LocalTileIdentifier == 9);
var animatedTile = (TiledMapTilesetAnimatedTile) myTile;
and once you've got the animated tile you can get access to the animation frames and so on...
All of the data you need is in there and you can get to it but of course it hasn't been designed for this use case yet so it's a bit fiddly.