Restrict by user's machine host name, possibly IP if she has a static one.
You can try Single Sign on with windows machines, that requires configuration on the machine.
Or simply set up the service on the singular machine, why does it need to be a website if only one computer can access it?
Single sign on varies in types from web single sign on (cross site authentication) to Machine Single Sign-on, which sounds like your preferred solution.
The good news is, If you develop this, your organisation end users needn't sign in to different applications, including web, and the authentication is that much safer.
The bad news is, it involves the Windows integrated sign on, and this means you have some control over the system sign in.
If this is not the case, and your users are not part of your organisation, you'll have to install a software solution on their machine, which can communicate with the Identity Provider.
For MS Windows Single sign on solutions you can see MSDN's security documents.
Let us know if this sounds like as a possible solution for you