Hi bmahin,
I'll take you through this from the start. The solution has been provided so I'm not adding new, just guiding through the stages.
Open Safari. Clear it of any residual Windows or Tabs.
Hold down Command and click on T (this Cmd + T option is just one of the ways of opening another Tab)
You should now see 2 Tabs at the top with "Untitled" on each of them.
Click on the Tab to your Left. Now go to the site where you are changing your password - as in your screenshot in Post #1.
Before requesting the change, click on the Right Tab and go to your wife's Gmail account so that you are ready for the 6-digit code.
Now back to the Left Tab. Request the change. Click on the Right Tab and await the 6-digit code.
When that appears, either write it down or use the Copy/Paste method if you are familiar with that.
Either way; back to the Left Tab and enter the code. Continue from there.
Hope that helps.
Ian