I think you just have to bind the Archived column to the checkbox. So line 39 will go from:
$UpdateQuery->bindColumn("Archived", "i", "1", "WA_IGNORE");
to:
$UpdateQuery->bindColumn("Archived", "i", "".((isset($_POST["archive"."_".$MultipleUpdateCount]))?$_POST["archive"."_".$MultipleUpdateCount]:"") ."", "WA_IGNORE");