Well, you can bet a defense attorney will try to get it in (eg. the Kobe case). But the judge shouldn't allow it.
But here's the interesting part. Let's say your Jenny Lee's braces guy committed a crime that, if found guilty, carried a mandatory minimum sentence of five years. Now what?
As a defense attorney, you don't want the jury to convict a guy trying to help his kid. The prosecutor doesn't want to have this kind of reputation. Neither does the judge.
Actually what would happen is what happens in drug cases. Knowing the braces story, and knowing the mandatory minimum, either the prosecutor would not bring the charge, or he would bring charges that didn't carry a mandatory minimun (say, shoplifting instead of robbery, I don't know).
What the mandatory minimum laws have done, therefore, is shift the penalty decision making from the judge to the prosecutor. And the prosecutor is a wheeler-dealer with an agenda, not an impartial judge of the facts.
Something to think about.