Apparently I need to read my console output from node-inspector more carefully. I was hitting my app by going to the URL:
http://localhost:8000/debug?port=5858
My console output from node-inspector was telling me to go through 127.0.0.1. I had initially ignored that, forgetting that node really does look at the URL and so node-inspector must as well. I was able to get to the debug console on chrome by going to this URL instead:
http://127.0.0.1:8000/debug?port=5858