Can't find what model Wii I have?

The model number/region is immediately to the left of the serial number on the sticker. (i.e.: "MODEL NO. RVL-001 (USA)")

It might be Korean.

 

It's very unlikely it can read burned discs. How did you end up with a HK Wii in the US? IMO, replace it with one from your region.

 

There are 4 regions of the Wii OS: Japanese, American, European, and later Korean (3DS did have a "Traditional Chinese" region sold in Taiwan, but that's a complicated story)

However nothing forbids a HK person or company from importing a Japanese console, optionally repacking it with localized manuals (as for the complicated 3DS story, the official Taiwanese model got so few releases and the kanji are written the same, that most HK gamers preferred Japanese models anyway; so did to a lesser extent the other 2 minor regions, China and Korea).
In fact the official Middle East consoles are rebranded American ones, and Australian ones European (another funny story: early Australian GC games had the European region lock but were based off the American version)

As for your console of unknown origin, it may well be a case swap - use a suitable homebrew to check the "real" serial number and region (which may still have been changed, but at least they will be the ones that matter)!

 

Does your console come with the shop channel and weather/news channels?
If it does not, then what you have is an official Hong Kong/Taiwan console.
However, there's not technically a HK/TW region for Wiis.Wiis from Hong Kong and Taiwan run on Japanese system menus and read Japanese-regioned games. All Chinese Wii games are also encoded in Japanese region so they can be played on HK/TW Wiis, as well as original Japanese Wiis.
The only difference is that the Wiis from Hong Kong & Taiwan do not have the Wii shop channel, forcast channel, and weather channels.
However the consoles are still compatible with WiiConnect24 and Nintendo Wi-Fi connection (Before the services ended, of course).

One interesting fact is that newer updates from Chinese/Japanese Wii games never install the channels onto a Wii from HK/TW, and shop/forcast/news channel WADs are actually found in update portion of Chinese Wii games.