Try any from below options:
$sql_query = "UPDATE pack_details SET $delivery_date = '{$_POST['delivery_date']}', $delivery_time = '{$_POST['delivery_time']}' WHERE $delivery_building = '{$_POST['delivery_building']}' AND $delivery_room = '{$_POST['delivery_room']}'";
or
$sql_query = "UPDATE pack_details SET delivery_date = '".$_POST["delivery_date"]."', delivery_time = '".$_POST["delivery_time"]."' WHERE delivery_building = '".$_POST["delivery_building"]."' AND delivery_room = '".$_POST["delivery_room"]."'";
Note: If field name doesn't contain $, remove $ from field name in query. For eg. "$delivery_date" should be "delivery_date"
Suggestion: Instead of using string concatenation for building, You should use bind parameters to pass value to query. It helps to prevent SQL injection as well as code look well.