= Solution
For steady <electromagnetic fields>, <Maxwell's equations> give $\nabla\times\boldsymbol B=\mu_0\boldsymbol j$ and $\nabla\cdot\boldsymbol B=0$. Introduce a <magnetic vector potential> with $\boldsymbol B=\nabla\times\boldsymbol A$ and the permitted <Coulomb gauge> $\nabla\cdot\boldsymbol A=0$. The vector identity for the double curl then gives $-\Delta\boldsymbol A=\mu_0\boldsymbol j$. For localized currents and the decaying solution,
$$
\boldsymbol A(\boldsymbol r)=\frac{\mu_0}{4\pi}\int_V\frac{\boldsymbol j(\boldsymbol r')}{|\boldsymbol r-\boldsymbol r'|}dV'.
$$
Here $-\Delta(1/(4\pi|\boldsymbol r|))=\delta$ follows from harmonicity away from zero and the inward flux of the radial derivative through a small sphere, whose magnitude is one. It verifies the vector Poisson solution componentwise. Moreover steady <electric charge conservation> gives $\nabla'\cdot\boldsymbol j=0$; integration by parts confirms the stated potential is divergence-free. Taking its curl, using $\nabla |\boldsymbol r-\boldsymbol r'|^{-1}=-(\boldsymbol r-\boldsymbol r')/|\boldsymbol r-\boldsymbol r'|^3$, proves the <Biot-Savart law>:
$$
\boxed{\boldsymbol B(\boldsymbol r)=\frac{\mu_0}{4\pi}\int_V\frac{\boldsymbol j(\boldsymbol r')\times(\boldsymbol r-\boldsymbol r')}{|\boldsymbol r-\boldsymbol r'|^3}dV'.}
$$
The zero-field condition at infinity selects the <magnetic field> produced by these currents, excluding an externally imposed homogeneous <magnetic field>.
At the origin write the source point as $r\boldsymbol e_r+z\boldsymbol e_z$. Its current is $kr\boldsymbol e_\phi$, so the numerator of the <Biot-Savart law> is $kr^2\boldsymbol e_z-krz\boldsymbol e_r$. The radial term cancels after angular integration. Using $dV'=r\,dr\,d\phi\,dz$ gives
$$
B_z(0)=\frac{\mu_0k}{2}\int_0^b r^3dr\int_{-h}^h\frac{dz}{(r^2+z^2)^{3/2}}.
$$
The inner <integral> is $2h/[r^2\sqrt{h^2+r^2}]$, as is verified by differentiating $z/[r^2\sqrt{r^2+z^2}]$. Hence
$$
\boxed{\boldsymbol B(0)=\mu_0kh\left(\sqrt{h^2+b^2}-h\right)\boldsymbol e_z.}
$$
For $b,h>0$, its magnitude is $\mu_0|k|h(\sqrt{h^2+b^2}-h)$; it points along $+z$ for positive azimuthal current $k>0$, and along $-z$ for $k<0$. The limit $h\to\infty$ is $\mu_0kb^2/2$, agreeing with direct integration of the infinitely long cylindrical current distribution.
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