Fixing clogged washer drain pipe
That snake is a nice one I would open that vent on the end and go straight in with the snake it should do the job, an augger only goes to the trap for the most part and in my experiance the plug is further down where other things connect. If you can't get that vent off you may be able to go to a sink that is close or find another clean out but remove the trap and go straight in. In my location that vent would require an air admittance valve since it is inside and some areas don't allow that at all (even though it is the same as a sink trap).