They don't contribute to society in any way except to kill mouses,
Mice.
And also, this is an understatement and you don't realize the importance of cats by that alone, so I stopped reading there and thought it would be better to explain why it's an understatement.
No animal was ever domesticated by mankind without purpose. Basically meaning, people needed the animal in one way or the other, so they domesticated them. If you look at history, at one point, agriculture was one of the main sources of livelihood for man
all over the world. Cats were domesticated because farmers realized that cats could kill two things that could kill crops, rodents & birds. Without cats in the picture, there would be nothing stopping mice, rats, and birds from raping every crop that every farmer planted. So this was basically a win win situation for both cat and man. The cat could stay near the farms where it could be assured that a lot of hunting opportunity would present itself. Man in turn would rely on the cat to kill off the rodents and birds, effectively resulting in a better harvest.
Come 14th century, cats were once against part of a reason mankind didn't entirely wipe out from the Bubonic plague outburst in Europe. The plague had killed off nearly 30 - 60% of Europe's population. It didn't affect the rest of the world as badly because Europeans in the 14th century would consider cats to be bad luck and killed off hundreds, thousands of cats, thinking they were associated with witchcraft. This led to outbreaks of the rat population which therefore hit Europe worse with the plague than any other part of the world. If cats hadn't been domesticated in other regions of the world, the plague would have had more devastating effects.
There are also other parts of history where cats have paid important roles for humans, so to say 'they don't contribute to society in any way except to kill mouses' is very shallow.
Let's not forget that no animal has been domesticated since probably 1500-1600, so the ones that were domesticated were therefore domesticated with reason. The fact that their original purpose of domestication may not be the main reason they are kept as pets these days doesn't mean they should be labeled as not contributing to society. They did play a part, and some of them still do