So, his urge to kill whoever sent him into the world he was in surged again; that's why he couldn't help but punch the mirror.
After punching the mirror, he just let the blood flow from his wounded fist to his wrist where the bracelet is.
According to the novel, the bracelet's space would be awakened after dropping blood on it, so now he just lets the blood flow from his fist to his wrist.
In the novel, Treyton didn't find the specialness of the bracelet at the beginning because he was skilled and wasn't injured.
He only discovered the space on his bracelet after he was injured in a fight and his blood flowed to the bracelet accidentally.
Now, he doesn't even need to find a reason to tell Treyton why he found the space of the bracelet because of punching the mirror.
As soon as his blood flowed to the bracelet, he could see that it was being absorbed.