(Please only read if you are not too squeamish.)
From personal experience, the humane traps are no so humane, either to the humans or the mice. The traps where you release the mouse outside only delay the mouse. They come back in the house. Believe me.
The ones where the mouse gets stuck on the trap are worse than the traps that snap the neck. The mice cannot get free, and you can't pull the mouse off without tearing skin and fur. The mouse will ultimately die a long death by starvation or infection from being removed (and there's still the possibility of it coming back in the house.)
We had a neighbor that used one of the sticky traps. I will say it wasn't good.
OP, you did the right thing getting a quick-death mouse trap. They are the only way to go. Mice are insidious creatures and multiply pretty rapidly. They must be kept under control. Otherwise, mouse droppings and urine end up all through the drawers and cabinets, in and on utensils and cooking tools. They can spread disease, too.
We used to live on the edge of a farm, and every spring and fall when the farm equipment when through, the mice would scurry into the house. Thankfully, we had two cats that are good at hunting, but we still had to use some quick-death traps to keep the mouse population under control. We now live in the woods, which means mice again. The cats do a good job, but when we first moved in, we had to use some traps.
I understand feeling of remorse, but I don't believe it is necessary. For your own safety, peace, and comfort, you needed to get rid of the mice. I believe you chose the best option for how to do so.