[in that day]: This will be after the Gentiles repent, as noted in the previous verse. It is also during the time when the record comes forth. Therefore, this statement is directed to the time during and after the coming forth of the Book of Mormon.
[then will I manifest unto them]: The Lord promises that faith will continue to be effective. Just as faith allowed the brother of Jared to see visions of the purposes of God (in addition to the person of God) so too will the saints in our day be able to receive these visions of the kingdom. Indeed, the process of the restoration of the church and the great doctrines of the kingdom that were revealed through Joseph Smith stand as a direct witness to the fulfilling of much of this promise. Even though Joseph Smith fulfilled it, we should not consider it completed. The promise is that faith will continue to be effective in communicating with God. Even though great things have been revealed, the promise is that the Lord stands ready to yet reveal more that we shall need, in the day that we shall need it. It is predicated upon our faith, like unto the brother of Jared.