11.6#9

I'm not sure what I'm doing wrong with this, so I'm probably doing it in
the wrong way altogether

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OK, first of all, you didn't tell me anything that you did to get to that point like you are supposed to.  Second, notice that you are engaging in a little psychological warfare against yourself? Stop that, it's harmful to yourself.  Webwork even told you got the first answer right, so why say that you're altogether wrong? Given that how far wrong could you be about the second question?

Using the arctan is the correct thing to do.  The argument of the arctan is not the ratio of the y and x directions though, since what this is talking about is the angle above the horizontal.  The horizontal in this case is any direction in the xy-plane.  You have to get the slope of z in the direction of the unit vector you use in the first part