Might be made known (gnōristhēi). First aorist passive subjunctive of gnōrizō with hina. The mystery was made known to Paul (Eph 3:3) and now he wants it blazoned forth to all powers (Gnostic aeons or what not).
Through the church (dia tēs ekklēsias). The wonderful body of Christ described in chapter Ephesians 2.
The manifold wisdom of God (hē polupoikilos sophia tou theou). Old and rare word, much-variegated, with many colours. Only here in N.T. Poikilos (variegated) is more common (Mat 4:24).