How not to kill the mouse in my house?
"Without wisdom, effort, restraint of the senses, without giving up everything, I see no well-being for beings."
  
  -- The Buddha (SN 2.17)


You can't have your cake and eat it too, unfortunately. The truth is the truth whether it is convenient or not. In this case, there are some fairly important truths that you have to consider when trying to decide what to do.

First is the truth that being a nuisance is a far lesser evil than murder in cold-blood... the mouse has the moral higher ground over the mouse-trapper. 

Second is the truth that you don't want to be murdered yourself; knowing this, how can you contemplate it for a fellow being?


  All tremble at violence; life is dear to all. 
  Putting oneself in the place of another, 
  one should not kill nor cause another to kill.
  
  -- The Buddha (Dhp. 130)


Third is the truth that killing makes you a killer; it changes the direction of your life in a significant way that will have potentially far-reaching consequences, far beyond feeling a little guilty for a little while. There are karmically significant mind-states involved in killing that may not be evident to you until you actually kill. Once you've begun to kill, you start to desensitize yourself to killing; it gets easier and easier, and your mind becomes further and further sullied. 

Fourth is the truth that the reason you can't sleep has little to do with the mouse and much to do with your own state of mind. In fact, lack of sleep itself isn't really the problem either; if your mind is clear, you can stay awake all night and not suffer harmful consequences.

My advice is first, to come to terms with your mind's reaction to noise that keeps you from finding peace; reminding yourself "hearing, hearing" is a good way to keep yourself objective and alert.


  “In that case, Bāhiya, you should train yourself thus:
  In what is seen there will be only what is seen,
  in what is heard there will be only what is heard,
  in what is sensed there will be only what is sensed,
  in what is cognized there will be only what is cognized.
  In this way, Bāhiya, you should train yourself.
  
  And since for you, Bāhiya, in what is seen there will be only what is seen,
  in what is heard there will be only what is heard,
  in what is sensed there will be only what is sensed,
  in what is cognized there will be only what is cognized,
  therefore, Bāhiya, you will not come to be because that;
  and since, Bāhiya, you will not come to be because that, therefore, Bāhiya, you will not be in regards to that;
  and since, Bāhiya, you will not be in regards to that, therefore, Bāhiya, you
  will not be here or hereafter or in between the two - just this is the end of suffering.”
  
  -- The Buddha (Ud. 10)


Second, consider whether it is your duty as a householder to take care of the mouse, or whether you can just as well leave it be. Unless it is destroying insulation or wiring, I don't think there is a compelling reason to remove a mouse; a rat, maybe... Either way, there are many ways of relocating a mouse humanly.

In regards to cats, if you in any way encourage a cat to catch and kill a mouse, well, it's not categorically different from killing it yourself...