Full Typekit walkthrough:https://ajar.freshdesk.com/support/solutions/articles/26000034920-using-typekit-fonts-with-indesign-in5-web-projects
Typekit is a bit tricky. Here's a better answer than my previous one...
You can attach the following as a .js file in the Resources section of the in5 dialog:
var typekitURL = "//use.typekit.net/REPLACE_WITH_YOUR_FILENAME.js";
var tkscript = document.createElement('script');
tkscript.src = typekitURL;
VERY IMPORTANT: The above code should get Typekit loaded, but you must also add the domains you're using to host your content to your kit on the Typekit website:
Note: Typekit will not work on your local machine (without a server), you must post your files to a server.
Also, you'll want to make sure the full font name (e.g. "Calluna Sans" rather than "calluna-sans") has been added to your kit, since that's how in5 will output it.
Here's a link with some other troubleshooting ideas.