Fragments of Your Soul by E.S. Erbsland

Since the sudden death of her father, Arvid Bergen is fighting to support herself and her unemployed mother. After she unintentionally crosses the border to another world, Arvid is driven by a single wish: to leave this dark mirror world as soon as possible. Looking for assistance with this seemingly impossible task, Arvid encounters the god Loke with whom she strikes a bargain. However, her confidence is suddenly shaken when she learns that Loke is a liar, traitor and murderer and one she should neither trust nor help. But longing for home, Arvid throws caution to the wind.