I was recently participating in a discussion about using poetry/songs as part of non-poetry-based stories. One primary example was the many people who complain about the poems/songs that are part of The Lord Of The Rings by J.R.R. Tolkien.
I generally do not enjoy seeing poems/songs in narrative fiction. That is mostly because the poems/songs are passive methods of integrating a backstory that could be accomplished more effectively via other means.
I'm not a big fan of poetry in general because poetry in an educational setting was presented as a decoding challenge where the reader is expected to have sufficient knowledge to understand obscure contextual references or "cute" phrasing. In my experience, the effort expended acquiring that knowledge rarely justified the experience of decoding the context or appreciating the "cute" phrasing. My Junior High humanities teacher always got the vapors over the poetry he used in class. I found those poems underwhelming.
As an example, I recall a poem that was ostensibly about the supposed beauty of lapis lazuli; a semi-precious stone. There were several pages containing stanza after stanza about a hunk of rock. Ostensibly, lapis lazuli was actually a subtext for something else. All the clues were there if the reader had sufficient knowledge to decode plain text to reveal the subtext.
It may be that the type of poetry matters in establishing a connection with the reader. I generally enjoy (and have collected) the poetry of e.e. cummings. I have a volume of Robert Frost's poetry at my bedside that I read and enjoy occasionally. And there are some epic poems (such as The Song of Roland) that I really got into.
Poetry is a form of expression with rules, traditions, and tropes that rarely connect with me. It is difficult for an author to make effective use of poetry/songs in actively moving the narrative of a larger non-poetic narrative forward. I've read the songs in the Lord Of The Rings. Those songs are (for me) an extended aside that does little to move the story along; a respite from an active story that offers passive illumination of characters.
That has generally been my experience with most poetry/songs that are included in larger, non-poetic tales.