Perhaps there is cooking grease gumming up the pipe. This would allow snake
to go through but retain water. Maybe hot water would flush it open?

They used to sell a garden hose attachment that swelled up as a ball
and was supposed to fit tightly in a pige. I don't know if you need
that or not.

I still see that stuff in hardware stores. You may also need to buy an
adapter to connect garden hose to the kitchen faucet.

If this still doesn't work, you could call a plumber with a drain pipe
camera, or use a pressure washer with a jetter hose.