I don't know if this really helps or not, but the best advice I can
give is...get (or rent) a better snake.

I rented one a while back. It used a heavier "snake" than what you
have and a heavy, flexible "sleeve" attached to the front of the spool
that you can run down into the pipe to prevent kinking in the most
likely place. I'm never going to waste my time with anything lighter
again...