I get the squick OP, but you need to stop being squirmy about it and sort it out! Is it def mouse not rat? Rat poo is about the size of a tic tac, mouse poo is more like a tiny string of sausages.
Firstly - calm it down, mice are gross but they're afraid of us and mostly only come out when they're sure no one is around. They keep to the edges of rooms/behind appliances etc because they have many predators and don't want to risk coming into the open.
I had a rat the other year and paid a pro about £200, it was gone in a day and luckily there was only one. The pro had a bait box that he put a snap trap into as well, so there was double danger in there! This is quite a good option as the bait boxes are child safe. If you're buying traps, get substantial ones, not the cheapest flimsy ones.
You can also get electric traps that electrocute the mouse when they step on a metal plate, less messy and kills outright. Plus again, they're child safe.
Peanut butter or choc works well, you only need a tiny bit. They can be cunning and have the bait faster than the trap snaps, I used tiny cut up bits of wine gum that you wedge onto the spike so they have to tug a bit and are more likely to get trapped.
Check your cupboards to see if they've got at any food. Keep food in sealed containers. Ask your neighbours if they've had any, if you're in a block of flats they could be elsewhere too. Do you rent? You might be able to get your landlord to help or the manager of the block of flats or even the council if it's an issue in the whole block.
Block any holes with steel wool. Up your cleaning game so you're not leaving food around, easier said than done with two small kids. I limited where kids are allowed to eat so it's easier to keep clean, eg no eating on sofa, only at table then sweep underneath.
Most fitted kitchens have cupboards with removable panels at the bottom down the floor - take these out and check for droppings. This is a good place to put snap traps/poison where your kids can't get them.