Complicated illustrator file need to understand how to clean up

A client provided me with a .ai illustration file of their drawing. The art was outlined some how..

Unfortunately, the paths are a mess. Bad hierarchy, outlines overlap not making connected shapes or fillable areas.

I really want to simplify so that it's a more workable file, but from what I can tell there is a lot going on, that makes it hard. I don't want to cut paths and reconnect them with the correct paths. The whole thing is a bit of a mess. A lot of the black outline is connected around a great deal of the drawing, but I feel as if the barbeque and fire should be a separate object from the apron and body area. Obviouslly hard to tell in looking at a .png file. I would love to tweak and make it cleaner.. Is this a situation having to do with trace and expand? or something else? Do I have to do this one tiny piece at a time? Sadly I can't get the original art.  Thoughts? Thanks.