My cat, whom I love and adore more than you is not "for rent." I seriously hope that you were joking when you asked to borrow him to take care of your mouse problem, because I find that highly offensive, not to mention disgusting.
How would you like it if I asked to borrow your dust-mop of a dog to clean the floors in my apartment? Bet you'd be pretty damn pissed.
Here are your options: A) Hire an exterminator B) Clean up your apartment C) Get a cat of your own D) All of the above