Do you have an in-sink food grinding garbage disposal thing?
I'd guess you've over used it (by trying to run some meat or heavy green vegetables down it), and it's created a clog down line from the washer pipe, and that's where the issue is.
Go outside near the kitchen, there should be an access opening that can be unscrewed... open it and flush it out as much as you can with a garden hose on full blast...
That should push the clog on down to the city sewer line...
If that cannot be done, or is not your case, then forget I mentioned it.
It worked for me. Most of the drains for wash rooms are "up line" from the same waste water drain that the kitchen sink is connected to, at least in my experience.
And if a plumber does it, it will cost much more than your time.