Let $l$ be distance of the point with the north pole and $x$ the distance it has gone at each time. Suppose it moves $\Delta x$ meters towards the northeast. if $\Delta x$ is small enough we can assume that it has moved in a plane. Then:\[ \displaystyle \Delta l=\sqrt{l^2+\Delta x^2-2l\Delta xcos(45)}-l \]. So $\lim_{\Delta x\rightarrow0}{\frac{\Delta l}{\Delta x}}=cos(45)=\frac{\sqrt{2}}{2}$. So after $\sqrt{2}l$ meters it reaches the north pole.It rotates infinate times around the north pole because we can compute$\lim_{\Delta x\rightarrow 0}{\frac{sin(\Delta \theta)}{\Delta x}=\frac{1}{\sqrt{2}l}}$.

Is it assumed that the earth is perfectly circular?

Yes, of course.