[Improvement] Rewrite in TypeScript

[theoludwig]

Type of Improvement

Refactoring code

Proposal

Rewrite the code in TypeScript.
The project wasn't started as TypeScript, as markdownlint types was light and often incorrect, but this changed, and greatly improved.
As the types improved, and are now "good", we use TypeScript to lint the JSDOC since this commit: 1ddcdc7.

As we're already using TypeScript, but to lint JavaScript, it makes sense, to instead directly use TypeScript syntax.