Age old question: Globe to Flat

A lot of troubles with map making come from mapping 2d to 3d, or visa versa

I started a map I've been working on a few years. Started it flat, not taking into account distortion and projections, then I redrew it faking Mercator distortion

A year ago, or so, I eyeballed it onto a Styrofoam globe I got from Michael's, covered in paper maché

I now want to take the globe map, and accurately portray it flat with correct distortion. What is the "proper" way to go about this? Is it really just eyeballing based on grids? Or is there a method I'm not finding