To the intent that (ινα). Final clause.

Might be made known (γνωρισθη). First aorist passive subjunctive of γνωριζω with ινα. The mystery was made known to Paul (Ephesians 3:3) and now he wants it blazoned forth to all powers (Gnostic aeons or what not).

Through the church (δια της εκκλησιας). The wonderful body of Christ described in chapter Ephesians 3:2.

The manifold wisdom of God (η πολυποικιλος σοφια του θεου). Old and rare word, much-variegated, with many colours. Only here in N.T. Ποικιλος (variegated) is more common (Matthew 4:24).