I am sorry that you are having to deal with a mouse in the house. I am sure that your feelings about killing are shared by many. I showed your letter to my wife and she reminded me of a time when she had a mouse cornered with a broom but could not bring herself to strike the shivering mouse with her "weapon." However, I want to assure you that it is alright to kill it in order to keep your house free of rodents. Let's consider our treatment of animals as taught in scripture.
First, the Bible teaches that righteous people treat animals kindly. Proverbs 12:10 states, "A righteous man regardeth the life of his beast: but the tender mercies of the wicked are cruel." This refers specifically to domesticated animals like farm animals and pets. A righteous man will regard the life of the animal while a wicked man will delight in cruelty.
On the other hand, the Lord allows the killing of animals for a couple of reasons. Animals may be hunted and killed for food (Genesis 27:5; Proverbs 12:27). In addition, dangerous or harmful animals can also be hunted and killed. Ezekiel 19:1-8 uses the capture of a dangerous lion as an illustration. I am sure that Solomon approved the extermination of the "little foxes, that spoil the vines" (Song of Solomon 2:15).
At the time of the creation of man, God blessed him and gave him "dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth" (Genesis 1:28). This authority is also a responsibility. We are to sensibly care for that part of creation that is under us, but we also have the right to make life and death decisions for that creation.
Now consider your mouse. It may not threaten your life in an external way. However, mice do bring diseases with them and they destroy property. You are allowed by God to kill it--not for the pleasure of killing--but for the protection of your household. Go ahead and use the poison. God makes a great distinction between the life of a man and the life of a beast (Matthew 6:26).
One final comment. Some might think me too soft because I recognize the good of having feelings for the animals. However, I think this is taught in scripture (Proverbs 12:10) and fear that we sometimes oppose something because those who deny the Bible believe it. As with all things, the Bible must be our authority for truth. God's creation is to be respected, not because it is our mother or our god, but because it is our responsibility and we are to be good stewards of this responsibility. May the Lord richly bless you and rid your home of mice.