When the washing machine drains the water backs up in the kitchen sink. But now the sink won't drain at all. My husband and son snaked out the drain hole out the kitchen window and got a lot of black gunk out, but it is still cloggrd. They even put the snake in the wasging machine drain hole. Nothing is working.

You have a blockage downstream from the kitchen sink. The solution is to snake the drain line from the sink's roof vent. Put out enough canle to reach the base and 20 foot more. Test by cycling the washer. Good luck, Tom