For $C^2$-embeddings you can follow weighted (length preserving) mean curvature flow until you have a circle, then move the center to 0, then use a homothethy to go to the unit circle, then use the contraction in $\{\phi\in Diff(S^1):\phi(1)=1\}$ to get arc-length parametrization, so you end up with a circle. The initial point of the parametrization is then the $S^1$ for your orientation. 

Now, how to get from topological embedding to $C^2$-embedding? Use convolution with a smooth bump function? Does this destroy topological embedding?