>>> How do I check if the wireless physical module is broken?
Respond = it's will simply dont detect anything of wireless Wi-fi, like in your case.
it's seem to be the Wi-fi hardware component inside your Wii console, please if you whant, re-talk with Nintendo about that possibility.
Then, ... This would be something you would have to speak to a Nintendo representative about. You can do so by calling Nintendo's support line at 1-800-255-3700. The support line is open from 6:00 A.M. to 7:00 P.M., Pacific time, every day.
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P.S. you can try Lan adaptor
https://store.nintendo.com/ng3/us/po/browse/productDetailColorSizePicker.jsp?productId=prod10279
Need to connect in the most reliable, fastest way? Don't have a Wireless Router? The Wii LAN Adapter is the solution!
The Wii LAN Adapter connects to an open USB port on the Wii U or Wii console and makes a direct connection to your broadband modem via Ethernet cable (not included) possible. This direct connection removes the risk of wireless network delays or issues when playing games online.
Not sure how to connect your Wii LAN Adapter, Ethernet cable, and broadband modem? We've got all the help you'll need here.
The Wii LAN Adapter is not required to connect your Wii U or Wii console to the internet if you want to connect via a wireless router. For information on setting up your connection to a wireless network, click here.