This sentence contains two conjoined subordinate clauses, both introduced by the subordinate conjunction that. In his copying from 𝓞 into 𝓟, Oliver Cowdery accidentally omitted the repeated that. The second that makes it clear that Nephi also spoke the second clause. Such examples of conjoined that-clauses are found throughout the text, as in this similar example:
For further discussion of the deletion of that, see under that in volume 3.
Summary: Restore the original repeated that in 1 Nephi 15:24 (“and that whoso would hearken”); such conjunctive repetitions are very common in the text.