Believe it or not, in my experience those ultrasonic pest repellers do work. I've got the Black & Decker brand (they come in multi-packs and don't break the bank), and they've been great. I've been using them for several years now. They don't bother my cat and the didn't bother my dog (may she rest in peace).

Please note, they don't pass through walls or solid objects, so you may need to set up a bunch of them at various angles/walls/rooms to cover the area needed. Also make sure you put one so that it covers your kitchen counter tops.

I had a similar problem in a place in San Francisco years ago. I had a very inquisitive, busy toddler and, obviously, didn't want to use anything toxic.

The first pest control company I called told me to buy a big bag of steel wool and get to work! I went through every nook and cranny in the apartment, stuffing around all the gaps between the wall and baseboards, around pipes under the kitchen and bathroom sinks, and installed new door sweeps to close off that gap under exterior doors.

I can tell you it worked immediately, without toxins or cruelty.

You could set it free in a park, if you have a large one somewhere around !

Would you be concerned about being humane if you knew the mouse was carrying a highly contagious diseases?

Get a cat. Also good for roaches.

My mom has a wound up mouse trap. Put a bit of food where it can not reach it from outside and must hop inside. It gets trapped. Then release outside of your apt (park?)

Saw these:

http://www.doyourownpestcontrol.com/info/rodents/multiple.htm#nonwindup

Google "Humane mouse trap" for videos on how to make a really simple mouse trap with a cardboard box, a toilet roll and some peanut butter. We had a really bad problem, but I used this trick and anytime I caught them, I took them out to park waaay far away and set them free to be food for the owls. Circle of life baby.

I worked at a 50 year old summer camp, cooking in an old log cabin kitchen out in the middle of the woods- we had our share of mice. I was interested in a humane solution, so I got some of those sonic plug-in mouse repellents, and they worked perfectly. Between that, and keeping my food stored in mouse-proof plastic containers, or the fridge, everything was fine. Didn't stop the rest of the staff from setting up traps and killing mice senselessly elsewhere... but yes, unless you are taking a mouse to a vet to be euthanized, there is no humane trap that kills mice. These things have startlingly high failure rates- they kill the mice eventually, but only after what can be hours of extreme suffering. I've seen if, and it's heart-wrenching. At the end of the day, it's our job to make our homes inhospitable to mice. They shouldn't be killed for our laziness. I really appreciate your sensitivity here.

The sonic repellers do work, but only for a while. Eventually, the pests get used to them and come back. Also, some humans (like me) can hear them and it's *really* annoying.
But they're good to drive out the pests and then do like @lella and plug every possible entry point (like holes around water pipes) with steel wool to keep them from coming back.

If you're going to capture and release, be sure to release it very far away. If it's close enough to come back in, it will.