I am not a plumber but am a general contractor and I am good with plumbing. How old is your house? Is the plumbing original? What size pipe runs to washer and kitchen? What kind of pipe is galvanized or ABS? But right off hand I would say it is clogged between the kitchen and where the bathroom plumbing ties in. The reason the the washer backed up first is because a washer pumps out the water alot faster than your kitchen sink can flow. And unless it is a food,S#$% back up that liquid drain cleaner doesn't do anything but just sit there. Euther snake it or cut out that section of pipe and replace it with a couplers. I have always just like replaceing the pipe if it is the original pumping and the house is allitle old. Any question just ask.