It sounds like, to some extent, what you are asking is whether an immersed closed curve in the plane can be extended to an immersion of a disk. This would yield the multiplicities that you seek; however, such an extension is not always possible. Necessary and sufficient conditions were first given in the thesis of Blank in 1967. For instance see [this paper][1] for more on this topic and references.


  [1]: https://arxiv.org/abs/1012.4923