I'm not a big fan of cats, but they are definitely the most efficient way to deal with mice
I wasn't aware you were supposed to be able to release the mice from the glue traps. Every time I've seen a glue trap used the mouse got his face stuck in the glue and suffocated, so I thought that was how it was supposed to work. My sister had one of those that 2 mice got stuck in at once and in their struggling around they got the electrical cord for the lamp stuck down in the glue underneath the mouse bodies. She was not at all impressed after having to untangle that mess.