  Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$
we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which
is sharp as $\alpha\to0$. We also show that the distributional determinant of
$u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a
by-product, for each $p>2$ we obtain some quantitative local
$W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for
$\infty$-harmonic functions in whole plane.
