WMI error 80041010 on performance counters

I recently was having problems with my WMI queries. Following some (bad) advice I rebuilt the repository. It didn’t solve my problem, and afterwards all the performance counter classes had disappeared. Win32_PerfRawData_* and Win32_PerfFormattedData_* were gone, reporting error 0×80041010 ["Invalid class"] (Instead of an error 0×80041010, MS says you might get error 0×80041002 ["Object could [...]