meta_query not working properly

Based on the codex, the meta_query parameter contains one or more array with the relation parameter not set if single inner meta_query array.

Also remove the page parameter as it serves only for a Static Front Page.

Your args array should look like that:

$archive_args = array(
    'post_type' => 'speight_home_plans',
    'orderby'=> 'title',
    'order' => 'ASC',
    'posts_per_page' => 12,
    'paged' => $paged,
    'meta_query' => array(
        array(
            'key' => 'display_where',
            'value' => 'speight',
            'compare' => 'LIKE'
        ), 
    ), 
);