Deathless by Julie Kagawa

How does one kill the unkillable?
Sparrow and her companions have been left with an impossible task: slay the immortal Deathless King, whose magic siphons the very life of the world to sustain himself.

The answer may lie deep in the past, when those who would become Deathless Kings stormed the sanctuary of the goddess of Fate, in an attempt to remove their threads from the Weave and become immortal. They broke off a piece of the goddess’s loom and hurled it to the earth, where it still lies at the center of a dangerous wasteland called the World Scar.

For any chance against the Deathless King, Sparrow and her companions must brave the World Scar and reach the loom, which would grant the power of a goddess. But the journey is fraught with danger, and the powerful Deathless King determined to find them.
And if the loom is reached—what will the power of a goddess do to a mere mortal? Sparrow has no choice but to find out.