Only 5 likely sources that I see (in no particular order), all of which call for a roofer:
1) a leak through the shingles uproof from the kitchen vent, and the water is running down either on the top of the tarpaper (roofing felt), or under it, and going down through the kitchen vent hole, if it is roughly directly downhill of the leak spot - which could be a roof leak or another vent/chimney uphill of it letting water in. Could also be same type of leak, running down the underside of the sheathing (it will do this on about 4:12 and steeper roofs) until it hits the first thing that blocks it - the kitchen vent duct, and runs down it.
2) leak in the flashing or caulking around the roof vent hood
3) rusted through roof vent hood, letting water in
4) a random roof leak, which just happens to start dripping down in that location
5) Could possibly be stuck flapper if the rains were real hard or blowing toward the flapper, and splash from rain hitting the exposed skirt of the vent splashed into the vent, then down along or into the duct.