In the episode "Control" a mother is on trial for the killing of a man who turned her daughter (and several other women) into a sex slave. During the trial, the daughter confesses on the witness stand. The DA moves for a mistrial, but the defense argues this would create "double jeopardy". This is completely untrue. Mistrials and re-trials happen quite frequently, and do not violate the double jeopardy clause.
The DA chooses to dismiss the charges against the mother. It later turns out that the mother really did murder the victim. However, the police state they can't arrest & charge her for the murder because (again) "double jeopardy" prevents it. This again is false. Since the charges were dismissed before closing arguments and jury deliberations, "jeopardy" never came into play. Therefore, refiling the charges & pursuing a new trial does not violate double jeopardy.