Can't Login to iCloud through System Prefs

My login Keychain account recently became corrupted, requiring that it be repaired, deleting its store of logins in the process, including those stored in the System Preferences under Internet Accounts. I was able to log into my Gmail without a hitch but iCloud is silently failing to log in. The UI does not change when this happens. I have logged in to iCloud.com and confirmed that the credentials are correct. I did successfully log in to iCloud once but, upon navigating away from Internet Accounts and quitting System Preferences, was immediately logged out.

This is particularly problematic because periodically I am prompted with sudden, related pop-up windows that read "com.apple.iCloudHelper.xpc wants to use the "login" keychain." or similar and which prompt me for my password. These seem like they would not crop up if my iCloud account were functioning.

I have attempted to repair my Keychain again but Keychain Aid assures me that there are (now) no problems with it.