DON'T put down poison if you like your cat or your neighbours like theirs! Your cat may not be very good at catching a healthy mouse running around but it might well eat a dead or dying one and then you have a risk of secondary poisoning. One poisoned mouse may not kill your cat (don't know if it would make it feel sick though) but several might... and where there is one mouse there are bound to be more (sorry!).
We had mice a while ago and the real solution is to fix any little holes they can use AND be scrupulously clean. If you can fit a biro in a gap, a mouse can get through it. We used mouse mesh on the outside air bricks, filled gaps under skirting, between floor boards, in the airing cupboard etc. Mice don't need to drink water to survive (they get enough from their food) and they only need a few grams of food per day so even a few crumbs can make a satisfying meal for them. It's very difficult with kids in the house but you just have to keep it clean. I was hoovering the whole house twice a day and the rooms the kids used three or four times! In your kitchen, make sure all food is stored in boxes the mice can't get into. We got some toy boxes with lids to use temporarily and it helped a lot. Try and keep things they could use for nesting boxed up too - old papers, fabric etc.
Traps might work but, as you've found, they can be difficult to get in the right place or some mice just know to avoid them. If you remove the food source, eventually they should go away (and block their access too). It can take a while and they may seem to be increasing in activity - when they get desperate, they start coming out at different hours to see what's around at different times. Freaky when you see them more but it's a good sign if you're seeing them during the day or rooms they wouldn't normally be in.