I did this on my current project. Basically:

- Set up your TypeScript config

- Set up Typings (or some other type def manager)

- Rename all ".js" files to ".ts"

- Replace all "var x = require()" calls to "import x = require()"

- Gradually refactor components (controllers, directives, services) to classes, or at least add type annotations to everything.

Because TypeScript is a superset of JavaScript, you can work on an untyped system and add gradual typing until the codebase starts looking nicer. Any new functionality can be implemented in TypeScript; any existing functionality can be slowly refactored when it is touched by changes.