Most scholars agree that 'that which is perfect' is when we put off mortality and put on immortality. When we are changed and glorified and enter into that eternal state of perfection promised to us by so many New Testament scriptures. (see examples below)
Paul is referring to the same thing here. When that happens we will know as we are known. That is when we will see him face to face. At the time this was written they used polished bronze mirrors and they were not perfect. Today we still know in part and it is obvious by the many interpretations that some have challenges in their understanding. Until then we have been given the Holy Spirit and the gifts of the Holy Spirit to help us not to usurp the authority of the scriptures but for mutual edification and we are waiting for that perfection when we will no longer need spiritual gifts.
F.F. Bruce also agrees with this interpretation. If you don't know who that is, you have not been serious about your biblical education. F.F. Bruce also states
"the literature of the period following the apostolic age makes it plain that the gifts did not come to a full stop with the closing of the New Testament canon."
That which is perfect is come is referring to the following:
53 For this perishable body must put on the imperishable, and this mortal body must put on immortality. 54 When the perishable puts on the imperishable, and the mortal puts on immortality, then shall come to pass the saying that is written:
“Death is swallowed up in victory.”
18 I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. 19 For the creation waits in eager expectation for the children of God to be revealed. 20 For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope 21 that[
a] the creation itself will be liberated from its bondage to decay and brought into the freedom and glory of the children of God.
22 We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. 23 Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for our adoption to sonship, the redemption of our bodies. 24 For in this hope we were saved. But hope that is seen is no hope at all. Who hopes for what they already have? 25 But if we hope for what we do not yet have, we wait for it patiently.
20 But our citizenship is in heaven. And we eagerly await a Savior from there, the Lord Jesus Christ, 21 who, by the power that enables him to bring everything under his control, will transform our lowly bodies so that they will be like his glorious body.
11 For the grace of God has appeared that offers salvation to all people. 12 It teaches us to say “No” to ungodliness and worldly passions, and to live self-controlled, upright and godly lives in this present age, 13 while we wait for the blessed hope—the appearing of the glory of our great God and Savior, Jesus Christ, 14 who gave himself for us to redeem us from all wickedness and to purify for himself a people that are his very own, eager to do what is good.
26 The last enemy to be destroyed is death. 27 For “God[
c] has put all things in subjection under his feet.” But when it says, “all things are put in subjection,” it is plain that he is excepted who put all things in subjection under him. 28 When all things are subjected to him, then the Son himself will also be subjected to him who put all things in subjection under him,
that God may be all in all.
So you see how you made up a story in your own mind about how the gifts ceased when neither scripture nor history supports your view and it is a view that is either ignorant of the historical documents or is willing to ignore them for a belligerent stubborn hostile stance against the gifts, not wanting them to be true and secretly wishing they had never been mentioned in the bible.