I would initiate several maneuvers, as follows:
Limit the time spent on videogames, and remove them altogether, if the rule is violated. You'll have to find a way to supervise their time.
I would require them to become involved in any or all of the following:
volunteer activity, sports, after-school special interest club, and also take a really good look at how they are performing academically. Perhaps some of that time and energy could be better devoted to schoolwork? Regulate times and days that they can do video games, and only after school work and chores are done. Step in and take charge! Do not be afraid of their suicidal suggestion, its probably just bullshit. But if you determine that they are in any way depressed or possibly serious, then get them to a professional, licensed counselor, pronto. Good wishes, good luck!