I was greeted with an error 204036 and given the option to try again, which gave the same results each time.
So, I tried to just download a copy of the channel wad (Netflix Channel v1024 USA) to install through a wad manager. This didn't work, either, as it kept giving me error -1022 (Content did not match hash in TMD).
I tried using a bunch of different IOSes to install, but nothing worked, though I ended up finding a solution:
Step 1. Go download a copy of the old Netflix Channel wad (NetflixChannel-v1-USA) and reinstall it via a wad manager.
Step 2. Once it's installed successfully, use the wad manager to uninstall it. (I got an error during uninstallation, but it finished okay anyway, FYI)
Step 3. Install the new Netflix Channel wad and it should complete successfully this time.
Apparently, the Wii's normal channel removal tool leaves behind some garbage that stops the new channel from installing properly. By reinstalling the old version and then uninstalling it through the wad manager, we can get rid of that garbage and end up with a clean slate.
Update: If this doesn't work for you right away, you may have to install IOS 56 and/or the Shop Channel v20 first. In my case, I'm still on a much older shop channel, but someone in the comments mentioned needing to do this before it would work.
Update2: It's asking for another update... If you follow the steps above, then uninstall Netflix via a wad manager, you should be able to download the updated Netflix channel from the Shop Channel.
Update: If this doesn't work for you right away, you may have to install IOS 56 and/or the Shop Channel v20 first. In my case, I'm still on a much older shop channel, but someone in the comments mentioned needing to do this before it would work.
Update2: It's asking for another update... If you follow the steps above, then uninstall Netflix via a wad manager, you should be able to download the updated Netflix channel from the Shop Channel.