Whilst your dropouts are BADLY chewed up, there's more than enough metal still there. The damage to your dropouts almost looks like someone tried to force it in the wrong way with a hammer.
Your problem is you're not lining them up right, they are not -- or at least should not -- be perfectly round. They have (or should have) two flat sides without thread that you line up with the fork. This is to help the shaft not rotate in the mounts.
Using a pair of slip-nose pliers or adjustable wrench hook onto the shaft, and slowly rotate it back and forth, and it should just drop in. A LOT of people miss the detail it's not fully threaded the whole way around. Sometimes they are smooth on one side only, and that usually goes towards the front of the bike. Other times they're smooth on two sides in which case it's reversible.
Yours looks like the former, but it's hard to tell from the pics. If you've got only one side smooth, rotate the shaft 180 degrees and try again.