Unable to Access Apple ID ("Phone number cannot be verified")

First, let me say thanks to anyone who can help. This is a longish read, and anyone who has the patience to get through it is a champ in my book.

Also, this problem doesn't quite relate directly to iCloud, but I didn't where else to post it.

I'm writing on behalf of my wife, who has a 16 GB iPhone 6s (bought about half a year ago in South Korea, where we have lived since 2008).

Short Version

As of this past Monday (December 19, 2016), my wife has been unable to sign into her Apple ID and thus cannot access anything in her iCloud or iTunes.

Long Version

As recently as Sunday (December 18, 2016), my wife's Apple ID was working fine. Just before bed, she had a notification to update iOS (I'm assuming to 10.2, though I'm unsure). Not wanting to bother with it just then, she ignored it and went to sleep. Next day, she woke to find that her iPhone had entered what looked to be Recovery Mode (image of lightning cable occupying the lower half of the screen, iTunes logo dead center, words along the lines of "itunes.com/restore" taking up the very top). She assumed that the phone had tried and failed to updated overnight, so she plugged it into our PC and opened iTunes.

In the iTunes device-summary page, there were two options: update and restore. She tried to update, and that failed. Then she restored, and that went fine. Then, of course, she had to go through the setup process--choosing language, enabling location services, choosing a Wi-Fi network. But at the Apple ID log-in step, things didn't work out. When she tried to sign in, the screen read "Phone number cannot be verified."

She turned the phone off, turned it back on, and then tried again. She was careful to go slowly, making sure to enter her password precisely. Yet again she got the message, "Phone number cannot be verified."

Soon, she had to go to work, and she left the phone with me, thinking maybe I could resolve it. For much of the morning, I got the same results as she did. At some point, I skipped the Apple ID log-in step and tried to log in later through Settings > iCloud. Well, that didn't work either. Same message: "Phone number cannot be verified."

Then I tried to log into her Apple ID via PC. I typed in the name and the password, clicked Enter, and then was taken to the Two-Factor Authentication page. And I went, "Hmm." She had not received a verification code, so I selected the Resend Code option. Still nothing came. Then I selected the Text Me option. Nothing came. Then I selected the Need Help > Start Account Recovery option. I went through the steps for account recovery, and therein I faced more obstacles. After inputting the Trusted Phone Number, I was eventually taken to the Recovery Phone Number page. It turned out that South Korea wasn't on the list of available countries, so I couldn't put in my phone number. Instead, I put in my mother-in-law's phone number. I clicked Continue only to receive a message on-screen that said "Too many codes sent." And that was peculiar. What codes?

It has been several days of doing much of the same thing, like houseflies tapping against a kitchen window.

Considerations

- There is no indication that anyone has tried to log into my wife's account illegally. She is not locked out of her ID; she simply cannot get into it.

- My wife has had her Apple ID since before the time she bought the iPhone, possibly before the time she moved to South Korea. She believes the phone number attached to her Apple ID is not the one she currently has (which is weird considering that the Trusted Phone Number is exactly the one she currently has).

Solutions Attempted

- Account recovery (detailed above)

- Generate Code in iCloud (unable to log in, we couldn't get to that option)

- Really anything even loosely related to the above two solutions

- Speaking with a friend who knows a great deal about Apple products

- Cursing a lot at inanimate objects

- Drinking to dull the anguish and frustration

That last one has had the most success, but "most success" in this case is no success at all.

As a temporary fix, my wife has set up a new Apple ID. This is less than ideal, of course. There is a wealth of photos, notes, and other stuff in her iCloud account that she wants back.

Thanks for reading, folks. And again, we thank you for any help you can give us.

Message was edited by: Gerund-DMC