Using the CI template jsduck and job mwext-jsduck-publish might just work, however https://www.mediawiki.org/wiki/Continuous_integration/Entry_points#JavaScript_documentation is inconsistent with that, see also T109005.
Regarding doxygen there is T66996.

jsduck is a ruby gems, we had a recentish version provided as a Debian package for Ubuntu Trusty but it turned out to be a nightmare to build for Debian Jessie. So instead we should move to use bundler to install it and define a rake doc command to generate it (Jan pointed to T109005).

Regarding Doxygen, other extensions have similar needs such as Echo T117025. Potentially one of the issue is generating the documentation would probably need a copy of MediaWiki core and it seems dumb to generate the whole mw doc for each extension.

Change 284475 had a related patch set uploaded (by Jonas Kress (WMDE)):
Add jsduck file for JS docs generation

https://gerrit.wikimedia.org/r/284475

Change 284477 had a related patch set uploaded (by Jonas Kress (WMDE)):
Add jsduck file for JS docs generation

https://gerrit.wikimedia.org/r/284477

I added the jsduck.json files to the relevant repos could someone please configure (or tell me how to do it) the CI to use them to build and upload the docs.

Change 284475 merged by jenkins-bot:
Add jsduck file for JS docs generation

https://gerrit.wikimedia.org/r/284475

Change 284477 merged by jenkins-bot:
Add jsduck file for JS docs generation

https://gerrit.wikimedia.org/r/284477

Kudos @Ladsgroup :]

Did I forget to mention @hashar? Would not have been possible without him. Thank you!