In my story a consortium of megacorporations spot a new comet 26km wide entering the inner solar system. Earth is in dire need of water because of an already worsening climate change disaster (think now + a hundred years of out of control capitalist greed). Especially the climate change disaster's effect on temperature, capitalist greed against inland aquifers and increasing salinity in coastal aquafers because of sea rise.
In countless places near Earth they are mining for water: the Moon and several near Earth asteroids. But this thing is huge, and timely, and in their greed, these megacorps think they can score a big win.
They try to bring the comet into orbit around the Moon with gravity tractors. In several years time they manage this, but the tractors didn't just bring a ball of ice, but a dirty ball of rock and ice - a rubble pile with thirty or so core rocks ranging from a few hundred meters across to a kilometer across. The gravity of the Earth and Moon finish pulling the thing apart and fourteen of the largest chunks of rock hit the Earth.
Given Earth is kind of a water world, most hit in the oceans, and throw hundreds of gigatons of water vapor into the air. The ones that hit land do the nastiest damage and a couple hit aquifers throwing not just ash and earth into the air, but also gigatons more water.
The mass of the atmosphere is around 5.3x10^15 gigatons, or 530 million gigatons, according to various sources.
The average amount of water in the atmosphere is 0.001% of all the water on Earth, or 1,386,000 cubic km, from the facts listed on the USGS's page on the water cycle. Wiki has a cubic meter of water as about a metric ton. So a cubic kilometer of water would be a gigaton. Given the previous facts, Earth's atmosphere should have about a 1.386 million gigatons of water in it.
I know greenhouse gas-wise, water vapor in the air is complicated. But because of all the dirt and ash in the air, first they'd have to go thru a prolonged nuclear/volcanic winter. After that, during that and for a long time after I expect it will rain like crazy. So the question is how long will it take for all those gigatons of ash and water vapor to come out of the atmosphere? The amount thrown up would only be few millionths of the total water in the atmosphere, but I would have to guess the suddenness of it would have some kind of a short term and long term effect.
Another complicating factor, and one I couldn't find any reference for is if a 1km size impactor would hit hard enough to cause any mantel plume. Various references say various things about how wide and deep a land crater will be for something that size.
Yet a third complicating factor - much of the rubble pile would still be big enough to cause significant air bursts and ground impacts, throwing up even more water and ash.
So given all that context - how long will the ash take to settle out? And how much longer will all that water vapor take to rain back down?