This answer is outdated. Also see my answer below.
This might not be the answer you hoped for. However, it might help some readers being new to JavaScript development.
I also was looking for documentation on the Jupyter (-Notebook) client API and could not find official docs.
The JavaScript source code can be found on the GitHub repository, for example:
What also helped was to
A. Use Google Chrome development tools, set a break point in the extension and type
Jupyter
in the console. Then you get a browsable tree structure that can be used to inspect the Jupyter object. That tree structure does not seem to contain methods.
B. Log all the properties of an object, including the methods. If you want to know about the methods of the notebook use for example
for(var property in Jupyter.notebook){ console.log(property)}
C. When typing in the console there is code completion. For example you might want to type
Jupyter.notebook.
and see the suggestions on how to complete the command:

Also see
Browsable tree structure in dev tools console:

How to list properties including methods:

...
