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/AdminLogsWindow.xaml

@@ -1,6 +1,6 @@
 <DefaultWindow xmlns="https://spacestation14.io"
             xmlns:logs="clr-namespace:Content.Client.Administration.UI.Logs"
             Title="{Loc admin-logs-title}"
-            SetSize="1000 400">
+            SetSize="1100 400">
     <logs:AdminLogsControl Name="Logs" Access="Public"/>
 </DefaultWindow>