The question of double jeopardy is not for a jury to consider. It is a question of law that is decided by the judge.
In this case, Rittenhouse is charged, among other counts, with first-degree reckless homicide. If he is found guilty on a lesser charge, he will have been acquitted of first-degree reckless homicide, and he will not be able to be retried on that charge by the state of Wisconsin. (If the trial is invalidated to the extent that jeopardy never "attached," for example if it is found that the judge had been bribed to prevent conviction, then a new trial could be brought.)
Once jeopardy attaches, there can be no additional trials for the same crime, and any subsequent attempts at prosecution should be dismissed by the court long before a jury is selected. By contrast, if a court determines that a trial does not constitute double jeopardy and a jury is seated, the jury will not consider the possible existence of double jeopardy; the only way to challenge a trial judge's decision on that question would be through the appeals process.
If the trial jury in the original trial cannot decide unanimously to acquit on a particular charge then it is a hung jury,and indeed that may result in a mistrial on only those counts on which the jury has deadlocked. From Wikipedia, quoting the Federal Rules of Criminal Procedure:
If the jury cannot agree on a verdict on one or more counts, the court may declare a mistrial on those counts. A hung jury does not imply either the defendant's guilt or innocence. The government may retry any defendant on any count on which the jury could not agree.
States, however, may approach this slightly differently.