Following output from sfc, I turned to read c:\Windows\Logs\CBS\CBS.log. At the time of failure, I saw the following lines:
Failed to find package: Package_78_for_KB2966828~31bf3856ad364e35~amd64~~6.3.1.4 from the index with mapping index packages recently rebuilt, [HRESULT = 0x800f090d - CBS_E_MISSING_PACKAGE_MAPPING_INDEX] Failed to get WU category/updateID for package: Package_78_for_KB2966828~31bf3856ad364e35~amd64~~6.3.1.4 [HRESULT = 0x800f090d - CBS_E_MISSING_PACKAGE_MAPPING_INDEX] Failed to get the category and update ID for package: Package_78_for_KB2966828~31bf3856ad364e35~amd64~~6.3.1.4, component: amd64_microsoft-windows-wpfcorecomp.resources_31bf3856ad364e35_6.3.9600.20708_en-us_a3b67b207d2057ab, file: PresentationHostDLL.dll.mui [HRESULT = 0x800f090d - CBS_E_MISSING_PACKAGE_MAPPING_INDEX] Exec: Missing files are found during staging attempt but failed to download content from WU. [HRESULT = 0x800f090d - CBS_E_MISSING_PACKAGE_MAPPING_INDEX] On a hunch, I uninstalled KB2966828 mentioned in the log, and voilà - installation works! I hope Windows Update will install it again if it's really important...