Finally got this sorted out.

For anyone else who runs into this, here's what the process seems to be:
1. Contact the operating airline (Air China) and send them your documentation (boarding pass etc).
2. Air China will send an email to United. They told me that the email address they use for resolving these issues is [email protected].
3. United should eventually get around to handling the email. For me, it took about 50 days.
4. Once it has been handled, the miles will be transferred to the correct account.

Step 3 is where I got really stuck. I called United several times to make sure they were being responsive to Air China's requests, but every agent I talked to gave me a different story. None of them were aware of the [email protected] email. Some of them also told me to just ask Air China to remove the miles from my Air China account and then resubmit my request for miles to United. This turned out not to be necessary once United finally got around to processing the email from Air China.