As per Paperback novel where a female kendōka kills an opponent because her sword is magically replaced, then she is transported to another world, The Swordswoman by Jessica Amanda Salmonson.
Kendo is a thing of the spirit, or so Erin Wyler thought of the ancient art of Japanese sword fighting--until the day in the fencing hall when her bamboo sword was transformed into a supernatural steel blade and her sparring partner became a loathsome demon. Suddenly kendo had become a thing of madness.
This review provides a few more details:
The story concerns Erin Wyler, a young Earth woman who's dreamt of living a hero's life in another world. She's a loner, going in and out of institutions her whole life and never finding any amount of happiness except in studying martial arts. During a kendo match, she kills her opponent but his wounds reflect a metal blade, not the bamboo stick she was using. An obsessed paramedic discovers that both she and her opponent have had dreams about the same world, only from opposite points of view. Erin dreams of being the rebel who kills an evil tyrant; her opponent dreamed of being that tyrant. In that alternate world, however, Erin's alter-ego Merilia is the one killed so forces present in both worlds conspire to bring Erin to Endsworld to fulfill Merilia's destiny.
Salmonson bases the alternate reality, Endsworld, on Asian influences. The world is ruled by a warrior elite whose ranks are differentiated by the number of swords they carry. One-swords are cannon fodder, minimal training with a short sword - the i. Two-swords are better trained, carrying both i and mai, and run the gamut from merely competent to expert. Three-swords carry the oude - a great sword - and are wuxia style masters.
The story follows Erin as she arrives in Endsworld and her development from unwilling pawn to someone able to replace Merilia.