I do not understand the question. Would you please elaborate what exactly a) does not work or b) you want to achieve.
Once a user has logged into the WordPress site on an Ubuntu server with credentials validated by the Active Directory Server over LDAP on the Windows servers, I want to send the user to a page on the Windows server website which requires credentials but I don’t want them to have to log in again. Is there some way to pass the credentials?
You could pass this as POST request parameter. But we will definitely not implement this. This is a security horror nightmare.
Kerberos is, what you want and should use.
A POST would work but as you point out isn’t very secure. There must be somew way to share credentials between the two systems for a single login.
As already written: With NADI we will not share any credentials. You may use Kerberos for SSO or you can use a solution like OAuth or Crowd to pass user information (but not credentials).