The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `company` ADD INDEX `company_idx_id` (`id`);
ALTER TABLE `houses` ADD INDEX `houses_idx_completed` (`completed`);
ALTER TABLE `landlord_houses` ADD INDEX `landlord_houses_idx_house_id` (`house_id`);
ALTER TABLE `tenant` ADD INDEX `tenant_idx_house_id` (`house_id`);

The optimized query:

SELECT
        DISTINCT h.id AS id,
        h.address1 AS address1,
        h.town AS town,
        h.postcode AS postcode,
        h.valid_from AS valid_from,
        h.valid_to AS valid_to,
        (CASE 
            WHEN c.name IS NOT NULL THEN c.name 
            OR ' (MS)' 
            WHEN h.letting_agent IS NOT NULL THEN h.letting_agent 
            WHEN t.id IS NOT NULL THEN t.letting_agent 
            ELSE 'Unknown (Not set yet)' END) AS agent 
FROM
houses h 
LEFT JOIN
landlord_houses lh 
    ON lh.house_id = h.id 
LEFT JOIN
company c 
    ON c.id = lh.company_id 
LEFT JOIN
tenant t 
    ON t.house_id = h.id 
WHERE
h.deleted IS FALSE 
AND h.archived IS FALSE 
AND h.sign_up_complete IS TRUE 
AND h.completed > NOW() - '14 days'::INTERVAL 
ORDER BY
h.id