@FreeDuck,
Oh hun - you're not a wuss. So sorry this happened to your kitty.
At my last house, our kitty was knocked over by a car - broke the kids hearts - we searched day and nite for her - Poppy - found her nearly a week later and had to bury her before the little fella came home, for the same reasons. All our kitties each had a grave ... and each had a carved name on a wooden plaque (PaPaIz carved them) and placed the kitties under a blossoming tree. The kids would often just go over and sit there in the paddock when they were feeling sad. Little fella still misses Poppy.
Let the kids know that your kitty wouldn't have felt any pain. Kids can't bear to think that their pet may have suffered at all. It was worse not knowing where Poppy was, when we found her, we all grieved for her then.
Have lost 3 cats to the road in years gone - and one who disappeared after we moved. Now have 2 kitties (one in the wars at the moment) - so completely understand how terrible you feel right now hun. Don't worry about crying with the kids either - that's part of the grief too... even for a cat/pet - they are part of your family.
Let us know how you go FreeDuck. Take care love. x