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.Shared/Administration/Logs/AdminLogsEuiState.cs

@@ -7,10 +7,11 @@ namespace Content.Shared.Administration.Logs;
 [Serializable, NetSerializable]
 public sealed class AdminLogsEuiState : EuiStateBase
 {
-    public AdminLogsEuiState(int roundId, Dictionary<Guid, string> players)
+    public AdminLogsEuiState(int roundId, Dictionary<Guid, string> players, int roundLogs)
     {
         RoundId = roundId;
         Players = players;
+        RoundLogs = roundLogs;
     }
 
     public bool IsLoading { get; set; }
@@ -18,6 +19,8 @@ public sealed class AdminLogsEuiState : EuiStateBase
     public int RoundId { get; }
 
     public Dictionary<Guid, string> Players { get; }
+
+    public int RoundLogs { get; }
 }
 
 public static class AdminLogsEuiMsg