Killing a mouse in the house with a third party
You are not responsible since the mouse came into your home upon its own volition. Also, you are not purposefully trying (wishing?) to create conditions in which a being would get killed.

But since you are aware of the situation, about a mouse and a cat roaming free inside your home. You should try to catch the mouse and then release it elsewhere in nature. In the meantime, you could try to prevent the cat from reaching this mouse.

The Buddha said: I declare that the mere arising of an intention of performing good deeds is productive of great benefits. - MN 8