I'm headed to Ohio later this month to help a school and parish out with a chanted Vespers for their patronal feast of Saint Michael, and just set the Litany for Saint Michael to music. My understanding is that the prayer has only been approved for private devotion, and there are several slightly different forms to be found in the bowels of the internet.

A couple slight adjustments:

1) "Christ hear us. Christ graciously hear us." was moved to the end of the prayer, where one would expect it in a chanted litany.

2) I changed the wording in the final prayer from "Pray for us, O glorious St Michael, Prince of the Church of Jesus Christ. That we may be made worthy of His promises." to "That we may be made worthy of the promises of Christ", as I expect some might think "His" refers to Saint Michael, rather than to Jesus Christ.

Feel free to use it! And if anyone has information about the origin of this litany, I'd love to hear about it.