She vowed to love him for the rest of her life.
I will never forget this kindness for as long as I live.
永遠 means 'eternity' in a more abstract or philosophical sense, whereas 末始終 is grounded in the span of a human life or a specific relationship.
Compound of 末 (すえ, 'end') + 始終 (しじゅう, 'from beginning to end'). The combination reinforces the idea of 'from now until the very end'.