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.Server/Administration/Logs/AdminLogManager.cs

@@ -374,4 +374,9 @@ public sealed partial class AdminLogManager : SharedAdminLogManager, IAdminLogMa
     {
         return Round(_currentRoundId);
     }
+
+    public Task<int> CountLogs(int round)
+    {
+        return _db.CountAdminLogs(round);
+    }
 }