HomePod mini, a year later and still broken

This seems like an issue you should have dealt with closer to the original purchase. Perhaps there is an issue with your Wi-Fi network that prevents proper configuration of the HomePod. You can run through Reset HomePod - Apple Support, in particular these steps:

If a white spinning light does not appear on the top of HomePod:

1. Unplug HomePod or unplug the power adapter for HomePod mini, wait 10 seconds, then plug HomePod back in.
2. Wait 10 seconds, then touch and hold your finger on the top of HomePod.
3. The white spinning light Keep your finger down.
4. Siri will say your HomePod is about to reset. When you hear three beeps, you can lift your finger.

After you’ve reset HomePod, you can set up HomePod

But they are unlikely to help if your network configuration is off. You need WPA2 and should not be using client isolation, guest mode, or similar settings that prevent peer-to-peer communications. Do you have your own Wi-Fi router where you are able to change its settings? Your phone should not be running any VPN or third party security software. Turn off private network address for your own network. See Use a private network address on iPhone - Apple Support for details.

You can also try Official Apple Support. It would have been better to press the issue with them while the device was new on the offchance that there is actually a fault that could have been rectified under warranty. Hopefully you'll be successful setting it up if you try again.

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