This is 99% of the way to a correct solution. But I think it would be better if pressing Escape would dismiss the help screen, rather than another press of F1,

This is 99% of the way to a correct solution. But I think it would be better if pressing Escape would dismiss the help screen, rather than another press of F1, so that the few people who intent to use help don't accidentally dismiss it. But nitpicks aside, this is the correct solution, because it captures the core insight: the user should never fear pressing the wrong key. The ultimate irony in violating this rule is when you've conditioned your users that the most dangerous thing they can do is ask for help.