Yes, it's fine.
In poetry, enjambment (/ɛnˈdʒæmbmənt/ or /ɛnˈdʒæmmənt/; from the French enjambement) is incomplete syntax at the end of a line; the meaning runs over from one poetic line to the next, without terminal punctuation. Lines without enjambment are end-stopped.
Basically, enjambment means to end the line with no or not-terminal punctuation before finishing the sentence on the next line.
So it is perfectly fine grammar to enjambment.
If you are specifically asking about the em dash in your enjambment, it is also fine because it is not terminal punctuation, meaning that the enjambment would still work.
You may consider switching out the em dash for a comma or semicolon if the em dash does not give the desired effect.
Personally, I like the em dash because it adds a more drawn-out feel and it adds a touch of drama to the continued sentence.
I understand it doesn't look elegant...
If the em dash is elegant or not is a mostly subjective topic but I would think that the em dash or a semicolon would be the most elegant. In the poem, the semicolon is already used so using it again would be more consistent.
Principally, if you are happy with the em dash as the end of your enjambment, then keep it.
If you ditch the whole enjambment thing and decide to end-stop, this is Wikipedia's definition for end-stopping:
An end-stopped line is a feature in poetry in which the syntactic unit (phrase, clause, or sentence) corresponds in length to the line. Its opposite is enjambment, where the sentence runs on into the next line. According to A. C. Bradley, "a line may be called 'end-stopped' when the sense, as well as the metre, would naturally make one pause at its close; 'run-on' when the mere sense would lead one to pass to the next line without any pause."
So to sum up, ending an unfinished poetic line with non-terminal punctuation (or no punctuation at all) is called an enjambment and it is grammatically correct for poetry.
Does this help?