If you can't be confident in your understanding of English being better than your daughter's, there's no hope for you. ;)
Joking aside, you're correct: for your daughter's version to be right, it would have to be "Please clean your toothbrush after you have used it." The form you've used, an example of an imperative, inherently implies "use it" as a continual, repeating action to which the instruction applies, so there's nothing wrong with that construction.
Always clean your toothbrush after you have used it.
Always clean your toothbrush after using it.
would be best, as the first one may confuse your daughter into thinking you only mean on that one occasion; it can mean that too. Nothing wrong with making it absolutely clear you mean for her to do it every time she brushes her teeth. :)
Edit: After some discussion on Language Overflow, I agree with one of the other users who pointed out that "after use" might be a bit too formal for a child, so I've changed my suggestion to "after using it".