The phrase "my soul delighteth" appeareth eleven times--sorry, appears eleven times--in the Book of Mormon, all in 2 Nephi. In each case, Nephi is the speaker.
It's important to note where in 2 Nephi the phrase does not appear: Chapters 1 thru the first half of 4 (where Lehi is the primary speaker), chapters 6, 9 and 10 (where Jacob is the speaker), and of course, chapters 7, 8 and 12-24 (the Isaiah chapters). However, in chapter 9, Jacob does say a couple of things very similar (though in the latter case, he's essentially just quoting Isaiah 55:2):
Behold, my soul abhorreth sin, and my heart delighteth in righteousness; and I will praise the holy name of my God. (2 Nephi 9:49, emphasis added)
...come unto the Holy One of Israel, and feast upon that which perisheth not, neither can be corrupted, and let your soul delight in fatness. (2 Nephi 9:51, emphasis added)I make note of this because, again, if the Book of Mormon is a fabrication, its authors at least deserve credit not only for creating countless characters with distinct voices, but also the ability to switch back and forth on a dime between those voices; the last words attributed to Lehi are in verse 11 of chapter 4, yet Nephi uses this phrase (for the first time) just four verses later, in fifteen, then again in the next verse. Additionally, when he closes his account of Jacob's discourse, Nephi immediately begins using the phrase again--repeatedly--starting in 2 Nephi 11:2.
(Side note: Given Nephi's proclivity for using this phrase in 2 Nephi, it's curious that he never does in 1 Nephi.)
"My soul delighteth" does appear twice elsewhere in the LDS Standard Works, and in both cases, the Lord Himself is the speaker, either through Isaiah (given Nephi's man-crush* on Isaiah, this is probably where he first picked it up) or Joseph Smith:
Behold my servant, whom I uphold; mine elect, in whom my soul delighteth; I have put my spirit upon him: he shall bring forth judgment to the Gentiles. (Isaiah 42:1)
For my soul delighteth in the song of the heart; yea, the song of the righteous is a prayer unto me, and it shall be answered with a blessing upon their heads. (Doctrine and Covenants 25:12)
Now, just for fun, here's a list of all the things Nephi's "soul delighteth in:"
the scriptures (4:15)
the things of the Lord (4:16)
[Isaiah's] words (11:2)
proving unto my people the truth of the coming of Christ (11:4)
the covenants of the Lord which he hath made to our fathers (11:5)
[the Lord's] grace, and in his justice, and power, and mercy in the great and eternal plan of deliverance from death (11:5)
proving unto my people that save Christ should come all men must perish (11:6)
plainness unto my people, that they may learn (25:4)
the words of Isaiah (25:5) (See? Man-crush.*)
to prophesy concerning him [the Savior] (25:13)
*The third definition of 'man-crush' on urbandictionary.com (a site I will not link to, this being a blog about holy writ): "A man having extreme admiration for another man, as though he wants to be him." (My man crushes, in case you were wondering: President Henry B Eyring, Michael J Fox, and this guy.)