I like the discussion [only possible in this dimension] which uses1] the fact that calderon zygmund operators which are smoothing of order one transforms  bounded measurable into continuous with elog1/e modulus of continuity and 2]  osgoods elementary theorem that this modulus of continuity is good enough for unique solutions of ODEs

Now in a chart with a bounded measurable change of almost complex structure thought of at each point as point in the unit disc with non-Euclidean geometry and the origin the standard structure in the chart draw the poincare geodesic between these two structures

applying the cauchy  transform converts the bounded measurable infinitesimal distortion of conformal structure into an osgood vector field we can integrate this and move along the geodesic path 
this produces a composition of homeomorphisms with the  required bounded conformal distortion 

 to get charts with holomorphic overlap mappings modify the starting charts by these constructed homeomorphisms with bounded conformal distortion  and use that one knows

 homeomorphisms with bounded conformal distortion and zero distortion ae are holomorphic


this proves newlander nirenberg for bounded measurable almost complex structures [relative to a standard background]

this is not the most elementary proof of  the smooth result  but to me it is the conceptually easiest proof 
and it gives a natural and  very strong statement with lots of applications not possible using the smooth result

Also the same proof scheme [use calderon-Zygmund then Osgood to inch your way to a solution]
also solves the euler equation for 2D incompressible fluid motion for any fixed time
which is not known in higher dimensions

Dennis Sullivan