A few points...
First, the prosecutor's question...it is not just a violation of the judge's ruling, it's a violation of Rittenhouse's Constitutional Right to Remain Silent under the 5th Amendment. I was a prosecutor, and that right there is one of the most basic rules of criminal trials...you cannot ask a defendant about remaining silent. Period.
Mistrial are not just a result of prosecution misconduct. Anything procedural that goes wrong during a trial that taints the jury's ability to fairly rule on a case can cause a mistrial. Often times, a mistrial can result from an honest mistake, so you basically start over with a new jury pool. An example I experienced was the judge, defense counsel and I discussing an evidentiary issue in the judge's chambers, not knowing the judge's wireless microphone had broken and was on (it would not turn off). The jury was in the courtroom and heard too much of the discussion before a deputy ignored the Do Not Enter sign on the office door and came in to tell us. Oops.
Now, if the judge declares a mistrial and dismissed the case with prejudice, it's done. The State cannot appeal, because it would amount to double jeopardy. That's why judge's very rarely dismiss with prejudice...fundamental fairness to both sides. But, given the way the judge ripped tge prosecutor's ass for this misconduct, I'll guess that it's possible but not likely. If there is a conviction, which I doubt, this will be on the appeal. And the judge knows this. And judge's don't like to have decisions reversed.