Is this something that never happened before? How is your laundry & kitchen positioned, does the kitchen sink and the laundry machine discharge into the same drain pipe / next to each other?
Did you try to use your plumbing snake line from the washing machine / kitchen side? The problem might be close to those two areas, not necessarily treatable from the exterior cleanout.
Do you have access (visual) to the plumbing pipes in the kitchen / laundry area or everything is hidden inside the floor / wall? If you can see it, describe the connections or (if possible) take a picture and email it to me.
Assuming that both (laundry & kitchen) areas are using the same plumbing vent pipe... it could be clogged with some debris, birds nest, beehive, etc. (unfortunately it would be accessible from the roof). However, it would most likely cause gurgling noises and water backup with smaller amounts of water as well; plug the sink, fill it up with water and remove the plug to see if it is draining without any problems.
Try to answer my questions to the best of your knowledge, just click “comment” below, thanks.