Thanks for all of the earlier replies with pictures, advice, suggestions, and feedback. A local FOG member (with the required equipment) has offered to help me make the repair.
The strain relief is indeed part of the cord. It cannot be moved. I did a quick search earlier on Amazon, but couldn't find a similar cord. I don't think I was searching for the correct terms - I was looking for "appliance cord with strain relief" but didn't have much luck. Amazon probably isn't the best place to try to find something like this, though I don't know where I should be looking.