Here is a short natural argument that planar continua are aspherical, different from the technique of Cannon/Conner/Zastrow, and straight forwardly applied to the Hawaiian earring.
The Hawaiian earring (and more generally any planar compactum $X$) is the nested intersection of planar polyhedra $X_n$.
The planar Euclidean metric naturally induces a length structure on $X_n$ so that $X_n$ is locally CAT(0).
In particular there is a natural proof $X_n$ is aspherical, since inessential loops in $X_n$ can be canonically extended to maps of the disk, and since the m-ball $B_{m}$ is naturally fibred by disks, so that the boundaries of the disks fibre the m-sphere $S_{m}$.
To see $X$ is aspherical take a map $f: S_{m} \rightarrow X$, and obtain the natural extensions $F_{n}: B_{m} \rightarrow X_{n}$.
Ascoli's theorem ensures the existence of subsequential limit and hence $X$ is aspherical.
In Cannon/Conner/Zastrow, inessential loops in $X$ are contracted internally, within their own image. In contrast, in the argument at hand, roughly speaking, inessential loops in $X$ bound disks fibred by limits of external geodesic chords.
To apply Ascoli's theorem we must gain global control of the equicontinuity data.
It sufficesis useful to show there exists a universal constant K>0 so that if P is a planar polyhedron, if $f: [0,1] \rightarrow P$ is a path, if C is the local geodesic path homotopic to f, (and if g is the parameterization of C induced by projection of f), then if (d,e) is uniform continuity data for f with respect to absolute value, then (d,Ke) is uniform continuity data for g (also with respect to absolute value). K=1.5 is apparently the worst case.
However, in the special case of the Hawaiian earring we may use K=1, since $X_n$ is the union of a finite bouquet of loops and a convex disk.