Let $\Pauli{\reg,t\magp}$ denoted the Pauli operator on a bounded open region $\reg\subset\R^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$.Assume that the corresponding magnetic field $B=\mathrm{Curl}A$ satisfies $B\in L\log L(\Omega)\cap C^\alpha(\Omega_0)$ where $\alpha>0$ and $\Omega_0$ is an open subset of $\Omega$ of full measure (note that, the Orlicz space $L\log L(\Omega)$ contains $L^p(\Omega)$ for any $p>1$). Let $\mathsf{N}_{\Omega,tA}(\lambda)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula\[\mathsf{N}_{\Omega,tA}(\lambda(t))\,=\,\frac{t}{2\pi}\int_{\Omega}\lvert{B(x)}\rvert\,d x\;+o(t)\]as $t\to+\infty$, whenever $\lambda(t)=Ce^{-ct^\sigma}$ for some $\sigma\in(0,1)$ and $c,C>0$.The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on $\mathbb{R}^2$.