As with other subordinate conjunctions in the original text of the Book of Mormon, since can be followed by that, although this is the only one in the text. In 18 other cases, the subordinate conjunction since is directly followed by the full clause—that is, without the that (as in the ubiquitous phrase “since the world began” found in 1 Nephi 3:20 and five other places). In his editing for the 1837 edition, Joseph Smith removed most instances of the archaic use of that after subordinate conjunctions, including the one following since here in 1 Nephi 22:5. For a complete discussion, see subordinate conjunctions in volume 3.
Summary: Restore the original that which followed the subordinate conjunction since in 1 Nephi 22:5.