What is in the Book of Mormon that fulfills the Bible?

This is a fun question. Unfortunately I only have time to mention a few things.

1. Jacob gave an interesting blessing to his son Joseph just before he died.
"Jacob is a fruitful bough, even a fruitful bough by a well; whose branches run over the wall.... The blessings of thy father have prevailed above the blessings of my progenitors unto the utmost bound of the everlasting hills...."
-Gen. 49:22-26
Often scriptures referencing fruit refer to posterity; e.g., "fruit of thy loins" (which to keep this simple I will quote again shortly).
The posterity of Joseph will run over the wall of the well. Obviously Joseph is not literally a branch, and he will not literally grow over the walls of a well. Branches (or tribes) of the house of Israel were lead away (or captured and enslaved) from time to time. the general scattering of the tribes is usually referred to as the "diaspora." -dispersion.
A few families heeded warnings from the Lord's prophets and fled before they were forced to.
Lehi and Ishmael were two such men who departed from Jerusalem with their families and were led across the ocean (WELL) to the ancient Americas around 600 BC.
Lehi was a descendant of Manasseh, who was the son of Joseph (Alma 10:3). These families had great blessings bestowed upon them and were highly favored of the Lord, thus fulfilling in part Joseph's blessing under the hands of Jacob.
The "everlasting hills" mentioned in this blessing, where some of the fruit of Joseph were to travel to, may be realistically identified as the mountain range stretching from the North tip of Alaska, to the southern tip of South America. -The Rockies and the Andes combine for this string of "everlasting hills."

2. Our friend previously quoted Ezekiel 37:16-19. Summarized briefly, two sticks are to be written on. One by Judah, one by Joseph. A stick is a scroll, much like the Dead Sea Scrolls. So two peoples are to write on scrolls and bring them together. The Bible contains mainly the writings of Jews (those from the tribe of Judah). The Book of Mormon contains mainly the writings of those of the posterity of Joseph (those who grew over the well). These TWO records combine to testify of the divinity of Jesus Christ and his supreme mission. "In the mouth of two or three witnesses shall every word be established." -2 Cor 13:1, Deut 19:15

3. John 10:16, is an interesting one also. “And other sheep I have which are not of this fold: them also I must bring, and they shall here my voice; and there shall be one fold and one shepherd.” Christ proclaims to the Nephites (descendants of Lehi-from the tribe of Joseph) that they and other scattered tribes are the “other sheep” he was to visit. See 3 Nephi 15:12-24.

Many nay-sayers have tried to make sense of this verse by saying that the other sheep were the gentiles. Here’s the problem, Christ specifically stated that these sheep were to hear HIS voice, and he also stated that HE was only sent to the lost (or other) sheep of the house of Israel (Matt 15:24). In fact the Apostles were not given the green light to preach the gospel to the gentiles until long after the ascension of the Savior when it was revealed to Peter in vision. See Acts10

I hope this helped.

Best wishes,
Rusty