I removed the addon from User Preferences/Addons. Once I did that I could no longer get the Blend4web addon to appear in the User Prefrences/Addon menu.
This is because Blender deletes addon files when you press "Remove". It's normal if an addon was installed from a zip file, but in case of "Preferences/File/Scripts" variant this operation is destructive and you need to restore the original files.
I have deleted blender preferences from User/appdata/roaming/blender. Reinstalled Blender and restored Blend4web SDK from back up zip file. Still can't get the Blend4web addon to show up in User Preferences/Addons. I must be missing something.
Can you think of any other files I can wipe to get things back to default?
This is strange, everything should work. Did you follow these installation instructions
? Also to make sure that the SDK isn't corrupted: are there addon files (like __init__.py) in the SDK/addons/blend4web folder?