Several questions need to be answered from 1 Corinthians 13:8-10. First, what is the partial? According to verse 9, the partial (meros) refers to the sign gifts such as prophecy, tongues, knowledge. As well, the term partial (meros) is something that is part of something more significant. For example, each believer is part (meros) of the body of Christ (cf. 1 Corinthians 12:27). In other words, prophecy, tongues, and knowledge are part of something bigger.
Second, what is that which is complete? It is not Jesus. The problem is with translation, not the text. The term perfect (teleios) refers to completion or maturity, not sinlessness. The term perfect (teleios) is a neuter noun; it refers to something, not someone.
That which is complete is the Scripture. The gifts of prophecy, tongues, and knowledge had a revelatory purpose: they were the means of imparting and confirming God's revelation to man. As noted, these gifts are partial (meros) or part of something bigger. Since these gifts are revelatory, they are part of the revelation (i.e., Scripture). If the gifts of prophecy, tongues, and knowledge are part of revelation, then that which is complete refers to the completed New Testament canon.
Third, when will it be established? Since the completed refers to the New Testament canon, it can be stated that it was established no later than AD 100. With the completion of the New Testament canon, tongues were discontinued, and prophecy and knowledge were rendered inactive.
What about the use of the gifts of prophecy and knowledge during the Millennial Kingdom? Just because the canon is complete now, does not mean that God cannot add to it later. Any addition will be authorized by Jesus and through prophets who meet the Deuteronomy 18 standard. This is why the term vanish away (katargeō) meaning to render idle or inactive, is in the passive voice. These gifts can be rendered inactive and then reactivated at a later time.