Fix "Next" never sending admin logs for rounds outside the cache, show a round's total logs on the UI (#16531)

* Fix next never sending logs for rounds outside the cache

* Show round's total log count on the ui

* Disable next button when waiting for a next response

* Cleanup AdminLogsEui.CurrentRoundId

* Fix popout window width

Content.Client/Administration/UI/Logs/AdminLogsEui.cs

@@ -70,6 +70,7 @@ public sealed class AdminLogsEui : BaseEui
 
     private void NextLogs()
     {
+        LogsControl.NextButton.Disabled = true;
         var request = new NextLogsRequest();
         SendMessage(request);
     }
@@ -88,7 +89,7 @@ public sealed class AdminLogsEui : BaseEui
             Maximized = false,
             Title = "Admin Logs",
             Monitor = monitor,
-            Width = 1000,
+            Width = 1100,
             Height = 400
         });
 
@@ -117,6 +118,7 @@ public sealed class AdminLogsEui : BaseEui
 
         LogsControl.SetCurrentRound(s.RoundId);
         LogsControl.SetPlayers(s.Players);
+        LogsControl.UpdateCount(round: s.RoundLogs);
 
         if (!FirstState)
         {