Hitting and killing any animal with your car is a terrible experience, and one that you will never forget, however, you have to remember that YOU are not the one who was at fault here! The cat's owner, if he even had one, is to blame for allowing his pet outside after dark where he could be killed or be badly hurt!
I think you were right to take your children home, and not expose them to the sight of the cat after it was hit, and you did return to the scene as soon as you could, to find the cat and see if it needed help or medical attention. After you found it dead, you also knocked on the door of the home nearest to where it happened, and inquired about its owner, and received no information.
Doing anything further than that, while you were alone and on a darkened road, would have been not only foolhardy, but dangerous as well. Your children need you, and putting yourself in harm's way for a situation that you couldn't do anything about would have made no sense at all! Your getting yourself mugged, or worse, certainly wouldn't have helped the cat, or its owner either!
If you had found the cat alive, but injured, then you should have taken it to the nearest vet, or called Animal Control or the Police in your area. However, since the cat was dead, moving it off the road, where it would not be hit again was really all you could do.
I hit a dog once, years ago, and I know how upsetting it can be, and how the experience haunts you afterwards. I know that it will be a long time before you stop being upset by what happened, but I truly believe that you did all you could, without putting yourself at risk.
Try to let it go, and be comforted by the fact that the cat probably died instantly, and never knew what hit him. He's in a better place now, where there's no more pain, and you are home with your children, safe, which is where you belong.