Some regular ol' Joe at the factory was able to fish that wire through there. I'm sure you can find a way too. Sometimes, it's just a matter of having the right tool.
Or the right technique. Don't just reef on your pull wire, but gently push on the replacement and gently pull on your pull wire. If it hangs up, back up a bit and twist the whole wire string. Electricians use a yellow goop for lubrication. but petroleum jelly or dishwashing liquid (maybe too thin to stay in place) is probably what you have.
Fusing that wire like Jerry said is also a good plan. There was obviously a sharp spot that wore thru the wire insulation on the originally. But if the fuse blows, you'll still need to find out why (and possibly replace the wire again.)
