Ezekiel 37:15-17 is about the then-future unification of the northern tribes of Israel with the southern tribes of Judah. The sticks represent the northern and southern kingdoms, not the Bible and the Book of Mormon. Let’s take a look at the context.

Ezekiel 37:15-17, “The word of the LORD came again to me saying, 16 “And you, son of man, take for yourself one stick and write on it, ‘For Judah and for the sons of Israel, his companions’; then take another stick and write on it, ‘For Joseph, the stick of Ephraim and all the house of Israel, his companions.’ 17 “Then join them for yourself one to another into one stick, that they may become one in your hand.”

Notice that verse 17 says that the son of man, Ezekiel, is to join them and they will become one in his hand.  Ezekiel goes on to write…

Ezekiel 37:21-22, “Say to them, ‘Thus says the Lord GOD, “Behold, I will take the sons of Israel from among the nations where they have gone, and I will gather them from every side and bring them into their own land;
22 and I will make them one nation in the land, on the mountains of Israel; and one king will be king for all of them; and they will no longer be two nations and no longer be divided into two kingdoms.”

God is prophesying through Ezekiel that he will gather his people who have been scattered throughout the nations and that he will bring them back to the land, and make them one kingdom again. Notice that verse 22 says “and one king will be king for all of them.”  It goes on to say who that king will be.

Ezekiel 37:24, “My servant David will be king over them, and they will all have one shepherd; and they will walk in My ordinances and keep My statutes and observe them.”

So, it should be obvious that the sticks which represent the northern and southern kingdoms, were to be united under King David. This has absolutely nothing to do with the Book of Mormon.

The LDS position

Generally speaking, the Mormon church does not examine the context of the lot of the citations that it uses in the Bible. Instead, the church picks and chooses those verses which will support its position.  Ezekiel 37: 15-17 is a good example of this misuse.  Let’s take a look at some LDS sources about this

• “The Prophet Joseph Smith in April 1830 identified the Book of Mormon as ‘the stick of Joseph in the hands of Ephraim (History of the Church 1:84.) And in August 1830, the book was again recognized as ‘the record of the stick of Ephraim.’ (D&C 27:5)” (Edward J. Brandt, “Using the New LDS Editions of Scripture – As One Book,” Ensign, October 1982, p. 42).
• “The stick or record of Judah—the Old Testament and the New Testament—and the stick or record of Ephraim—the Book of Mormon, which is another testament of Jesus Christ—are now woven together in such a way that as you pore over one you are drawn to the other; as you learn from one you are enlightened by the other. They are indeed one in our hands. Ezekiel’s prophecy now stands fulfilled” (Boyd Packer, “Scriptures,” Ensign (Conference Edition), November 1982, p. 53).
• “But Isaiah was not the only Old Testament prophet who foretold the Book of Mormon. Ezekiel wrote: ‘Take thee one stick, and write upon it, For Judah, and for the children of Israel . . . : then take another stick, and write upon it, For Joseph, the stick of Ephraim, and for all the house of Israel . . . : And join them one to another into one stick; and they shall become one in thine hand.’ Today, Saints living in many nations of the earth gratefully hold the Bible (the stick of Judah) and the Book of Mormon (the stick of Ephraim) bound as one in their hands” (Russell M. Nelson, “Scriptural Witnesses,” Ensign (Conference Edition), November 2007, p. 44. Ellipses in original).

What I find interesting is that the eighth article of the Mormon church says, “We believe the Bible to be the word of God as far as it is translated correctly; we also believe the Book of Mormon to be the word of God.”  The general implication held by Mormons is that the Bible is not as trustworthy as the Latter-day Saints Scriptures include in the Book of Mormon, the Doctrine and Covenants, and the Pearl of Great Price.  Yet, the Church of Jesus Christ of Latter-day Saints will still select verses out of the Bible when it suits its need.