We analyze a square-lattice random-bond Ising model using a numerical transfer-matrix technique to test the theory proposed by Lipowsky and Fisher for the complete wetting transition of a wall by one of two phases which coexist in a random medium. The theoretical scaling arguments are checked in detail. The transverse and longitudinal correlation lengths, ${\ensuremath{\xi}}_{\ensuremath{\perp}}$(h) and ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$(h), are found to be related by ${\ensuremath{\xi}}_{\ensuremath{\perp}}$\ensuremath{\sim}${\ensuremath{\xi}}_{\ensuremath{\parallel}}^{\ensuremath{\zeta}}$ with \ensuremath{\zeta}=0.65\ifmmode\pm\else\textpm\fi{}0.02 as the external field, or chemical potential deviation, h, approaches 0: the theoretical expectation is \ensuremath{\zeta}=(2/3. The mean wetting layer thickness diverges as l\ifmmode\bar\else\textasciimacron\fi{}(h)\ensuremath{\sim}${h}^{\mathrm{\ensuremath{-}}\ensuremath{\psi}}$ while as h\ensuremath{\rightarrow}0 with \ensuremath{\psi}=${\ensuremath{\nu}}_{\ensuremath{\perp}}$=(1/2. .AE