Windows Update error code 0x80240439 and web service xml errors

I'm running Windows 10 Pro, Version 1511 (Build 10586.545). I wanted to install the Xbox Accessories App, but got the error code 0x80240439 without any further details. As it turns out, I get that error when I try to install any app (which definitely did work at some point before).

I tried to investigate and found this in the Windows Update logs (a few lines are in german, sorry, but the gist should be understandable):

2016.09.08 00:08:55.2999858 676   792   WebServices     Auto proxy settings for this web service call.
2016.09.08 00:08:55.6828367 676   792   WebServices     WS error: Fehler beim Vorverarbeiten der Eingabequelle durch den XML-Reader in Zeile 1, Spalte 1300 (0x514).
2016.09.08 00:08:55.6828370 676   792   WebServices     WS error: Ein Endelement wurde erwartet.
2016.09.08 00:08:55.6828373 676   792   WebServices     WS error: Ein unerwartetes Startelement mit dem Namen CacheLifetimeMinutes" und dem Namespace "http://www.microsoft.com/SoftwareDistribution/Server/ClientWebService" wurde gefunden."
2016.09.08 00:08:55.6828381 676   792   WebServices     Web service call failed with hr = 80240439.
2016.09.08 00:08:55.6828381 676   792   WebServices     Current service auth scheme=0.
2016.09.08 00:08:55.6828384 676   792   WebServices     Current Proxy auth scheme=0.
2016.09.08 00:08:55.6836675 676   792   SLS             Retrieving SLS response from server using ETAG UODGUpPpKUSECGXeqbwPMdg0BYySCWk9UJebGlbHGPI=_1440"..."
2016.09.08 00:08:55.6837677 676   792   SLS             Making request with URL HTTPS://sls.update.microsoft.com/SLS/{855E8A7C-ECB4-4CA3-B045-1DFA50104289}/x64/10.0.10586.0/0?CH=739&L=de-DE&P=&PT=0x30&WUA=10.0.10586.545
2016.09.08 00:08:56.6741275 676   792   Misc            StatusCode for transaction returned from WinHttpQueryHeaders is 304
2016.09.08 00:08:56.6766369 676   792   Misc            Got 855E8A7C-ECB4-4CA3-B045-1DFA50104289 redir Client/Server URL: https://fe3.delivery.mp.microsoft.com/ClientWebService/client.asmx""
2016.09.08 00:08:56.6777986 676   792   Misc            Token Requested with 0 category IDs.
2016.09.08 00:08:56.6927945 676   792   Misc            Acquired new token from Server
2016.09.08 00:08:56.6928651 676   792   Misc            Got service 855E8A7C-ECB4-4CA3-B045-1DFA50104289 plugin Client/Server auth token of type 0x00000001
2016.09.08 00:08:56.6932602 676   792   ProtocolTalker  PTError: 0x80240439
2016.09.08 00:08:56.6932613 676   792   ProtocolTalker  SyncUpdates_WithRecovery failed. 0x80240439
2016.09.08 00:08:56.6932642 676   792   ProtocolTalker  SyncUpdates round trips: 1
2016.09.08 00:08:56.6932644 676   792   ProtocolTalker  Sync of Updates 0x80240439
2016.09.08 00:08:56.6932667 676   792   ProtocolTalker  SyncServerUpdatesInternal failed 0x80240439
2016.09.08 00:08:56.6936103 676   792   Agent           Failed to synchronize, error = 0x80240439
2016.09.08 00:08:56.6962830 676   792   Agent           Exit code = 0x80240439
2016.09.08 00:08:56.6962835 676   792   Agent           * END * Finding updates CallerId = WSAcquisition  Id = 23
2016.09.08 00:08:56.6987966 6852  4772  ComApi          *RESUMED* Search ClientId = WSAcquisition
2016.09.08 00:08:56.6990560 6852  4772  ComApi          Updates found = 0
2016.09.08 00:08:56.6990563 6852  4772  ComApi          Exit code = 0x00000000, Result code = 0x80240439

The first few lines sound like the XML response from a web service is invalid. How could that happen? I doubt that the Microsoft servers are returning incorrect data, so it's probably something on my device?

Apparently my Windows Update was broken in some still unknown way, and didn't update to the most recent Windows version. As Ramhound suggested, I manually updated by running the executable from http://go.microsoft.com/fwlink/?LinkID=799445, and then it worked.