if I understand correctly you need a 'group by' in your query; 
so like
SELECT  day("dtmDateCompleted") as "Day", 
        "A"."strName" AS "Category", 
        COUNT(*)
  FROM "tblAssetCategory" "AC" 
       INNER JOIN "tblAsset" "A" ON "AC"."id" = "A"."intCategoryID"
       INNER JOIN "tblWorkOrder" "W" ON "W"."intAssetID" = "A"."id"
 WHERE "dtmDateCompleted" IS NOT NULL 
GROUP BY day("dtmDateCompleted"), "A"."strName"



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