I agree with Jakob Nielsen that user should simply check what password they type. But if they want to. That solves many login issues.
I took latest jQuery 1.4.2 and Prototype 1.6.1 with 4 modern browsers for tests: Google Chrome, Mozilla Firefox, Opera and Internet Explorer 8.
Good looking solution not works in IE because it doesn't let you to change DOM's element attribute 'type" once it cloned.
But if you still concerned that IE users are good people, just got it preinstalled with OS - I have an ugly tricks version that works in all these browsers.
One thing, I forgot to mention.
When adding Show feature, you braking tab key flow for those, who like to use primary a keyboard. So now, after Password field, pressing Tab move us to Show checkbox, but it should to Remember me as would without it. In this case, you have to set tabindex property manually, to keep user on track and not show a secret incidentally to people looking at monitor instead of checking Remember me. See example below.