TASK 9.4: live chat wiring — right panel with Super Chat + membership badges

- YouTubeStreamService: added GetBroadcastLiveChatIdAsync to fetch liveChatId from broadcast contentDetails
- YouTubeChatService.Poll(): parse snippet.type to handle textMessageEvent, superChatEvent, newSponsorEvent
- MainViewModel: wire _youtubeChat.Start(liveChatId) in PrepareAndStartLiveAsync after broadcast creation
- MainViewModel: wire _youtubeChat.Stop() in StopStream before encoder stop
- MainViewModel: wire _youtubeChat.Dispose() in Shutdown
- MainWindow.xaml: updated chat DataTemplate with member badge (⭐), Super Chat highlight (💰 + gold border), timestamps (HH:mm)
- MainWindow.xaml.cs: auto-scroll ChatListBox on new messages via CollectionChanged
- Chat lifecycle is non-fatal: failures log and the stream continues without on-panel chat

Services/YouTubeStreamService.cs

@@ -190,4 +190,28 @@ public class YouTubeStreamService
         }
         return health;
     }
+
+    /// <summary>Returns the liveChatId for a broadcast by fetching its contentDetails.
+    /// The liveChatId is needed to poll chat messages. Called after broadcast creation
+    /// in PrepareAndStartLiveAsync.</summary>
+    public async Task<string?> GetBroadcastLiveChatIdAsync(string broadcastId)
+    {
+        if (!await EnsureToken()) return null;
+
+        _http.DefaultRequestHeaders.Authorization = new("Bearer", _auth.CurrentChannel!.AccessToken);
+        var response = await _http.GetAsync(
+            $"{ApiBase}/liveBroadcasts?part=contentDetails&id={broadcastId}");
+        if (!response.IsSuccessStatusCode) return null;
+
+        var json = await response.Content.ReadAsStringAsync();
+        var data = JsonSerializer.Deserialize<JsonElement>(json);
+        var items = data.GetProperty("items");
+        if (items.GetArrayLength() == 0) return null;
+
+        var contentDetails = items[0].GetProperty("contentDetails");
+        if (contentDetails.TryGetProperty("liveChatId", out var liveChatId))
+            return liveChatId.GetString();
+
+        return null;
+    }
 }