WSUS Cannot Sync with Microsoft Update

WSUS was installed on a Windows 2008 R2 server in our network. It stopped working a few months ago, and now the computers in the domain can't get any updates, either from the WSUS server or from Microsoft Update directly.

I just installed WSUS on a Windows 2012 R2 server, and am trying to set it up. It can't sync with Microsoft Update either. Here's the error message.

Synchronization error details:
A HTTP error occured:

WebException: The underlying connection was closed: An unexpected error occurred on a send. ---> System.IO.IOException: Unable to read data from the transport connection: A connection attempt failed because the connected party did not properly respond after a period of time, or established connection failed because connected host has failed to respond.
at System.Web.Services.Protocols.WebClientProtocol.GetWebResponse(WebRequest request)
   at System.Web.Services.Protocols.HttpWebClientProtocol.GetWebResponse(WebRequest request)
   at Microsoft.UpdateServices.ServerSync.ServerSyncCompressionProxy.GetWebResponse(WebRequest webRequest)
   at System.Web.Services.Protocols.SoapHttpClientProtocol.Invoke(String methodName, Object[] parameters)
   at Microsoft.UpdateServices.ServerSyncWebServices.ServerSync.ServerSyncProxy.GetAuthConfig()
   at Microsoft.UpdateServices.ServerSync.ServerSyncLib.InternetGetServerAuthConfig(ServerSyncProxy proxy, WebServiceCommunicationHelper webServiceHelper)
   at Microsoft.UpdateServices.ServerSync.ServerSyncLib.Authenticate(AuthorizationManager authorizationManager, Boolean checkExpiration, ServerSyncProxy proxy, Cookie cookie, WebServiceCommunicationHelper webServiceHelper)
   at Microsoft.UpdateServices.ServerSync.CatalogSyncAgentCore.SyncConfigUpdatesFromUSS()
   at Microsoft.UpdateServices.Serve