Beautiful riddle! Though I really don't know the answer
Hmm... maybe Nimrodel?
By falling of my namesake did I dwell, - the waterfall of Nimrodel
Till from the mountains came a fell. - the Orcs in the Mountains
Then promise was made to seal my fate, - she went to the south with Amroth
Yet in the end I came too late. - Amroth's ship was already away when she came there
Stubborn in ways, and bane of King, - "the King" is Amroth, of course: an elven king
Listen close, for still I sing. - it's said her voice still sings there.