“Where… like” query not working as expected

You should definitely be using prepare in this case, as your query is accepting input. It would be quite trivial for someone to compromise your database otherwise. Here we also use the new-in-4.0 esc_like method:

$like="%" . $mydb->esc_like( $facility_name ) . '%';
$results = $mydb->get_results(
    $mydb->prepare(
        "SELECT facility_name FROM facility WHERE facility_name LIKE %s LIMIT 0, 10",
        $like
    )
);