KB2898849 breaks HTTPS communication with one particular Web site

After installing the KB2898849 patch (Update for Disabling RC4 in .NET TLS) on a Windows 8 machine, the following .NET 4.0 code consistently times out while waiting for a response from one particular IIS server. (The code works with other servers.)

string url = "https://...";
WebRequest request = WebRequest.Create(url);
request.Method = "POST";

using (var stream = request.GetRequestStream())
using (var writer = new StreamWriter(stream))
{
	writer.Write(requestXml);
}

using (var response = request.GetResponse())
using (var stream = response.GetResponseStream())
using (var reader = new StreamReader(stream))
{
	responseXml = reader.ReadToEnd();
}

The exception is "System.Net.WebException: The operation has timed out", thrown by System.Net.HttpWebRequest.GetResponse().

With the patch, the System.Net trace log shows the following:

ProcessAuthentication(Protocol=Tls12, Cipher=Aes128 128 bit strength, Hash=Sha1 160 bit strength, Key Exchange=RsaKeyX 2048 bit strength)

Without the patch, the trace log shows the following:

ProcessAuthentication(Protocol=Tls, Cipher=Rc4 128 bit strength, Hash=Sha1 160 bit strength, Key Exchange=RsaKeyX 2048 bit strength).

How can I troubleshoot this issue further?