JetBrains: Cover TypeScript sources in CI

Created by: philipp-spiess

Fixes #35787 (closed)

This PR adds support for verifying the correctness of TypeScript files inside client/jetbrains so that we avoid issues like this in the future.

Since we do not have a test suite yet, we only verify that the extension builds properly.

Additionally, I've added client/jetbrains to the list of all ts projects so that things like our linter is set up correctly.

Test plan

• GitHub tests