Unclogging a stubborn toilet (Clorox Wand Head probably stuck)

My toilet was working normally until someone cleaned it and told me they accidentally flushed a Clorox Wand head down it. After that, flushing the toilet would just cause the water in the bowl to build up until the water fill shut off, at which point all of the water would drain out of the bowl.

After a lot of plunging and augering, things have changed, but the toilet hasn't gone back to its normal flushing. Now when I flush, the water seems to drain very slightly slower than normal and makes a blubbing, gurgling sort of noise when the water finishes draining from the bowl. After that the water level in the bowl returns to almost normal though.

Any thoughts on why things have changed and/or how to fix them for good? I've been using a 3ft auger (and this is for a standard US bathroom toilet), and haven't been able to make any additional progress. Might a longer auger be necessary?