Translation:I would like to be the reason for your sleepless night.
Lol. Often people associate falling in love with having sleepless nights in the sense that you stay up thinking about the one you love or because of butterflies in your stomach because you are in love and all the romantic crap. So, the person wants to be the reason for those sleepless nights. His/her lover, in other words.
Yes, it's indeed wrong. Nacht is the singular, Nächte the plural. In the plural, the adjective also has the ending -en, so the full sentence would be:
"Ich würde gerne der Grund für deine schlaflosen Nächte sein."
Edit: Deleted the second paragraph, I'm not so sure how correct it was what I wrote there and I don't wanna spread misinformation