jmc

Let $x=\angle BAC$. Angles $\angle BAC$, $\angle BCD$, and $\angle CBD$ all intercept the same circular arc, minor arc $BC$ with measure $2\angle BAC=2x$. Then $\angle BCD=\angle CBD=x$, so $\angle D=\pi -2x$.

Since $\angle ABC=\angle ACB$, $\angle ABC=(\pi -x)/2$. Then from the equation $\angle ABC=2\angle D$, $$\frac{\pi -x}{2}=2(\pi -2x).$$Solving for $x$, we find $x=3\pi /7$, so $k=\boxed{3/7}$.