The extender – be it an wired access point or a wifi repeater – just feeds what it gets from the router.
I had 4 kids at home (4500 sq feet) and had the same problem. There are many ways to tackle this problem but you absolutely need it to be easy for you. Get the best router you can buy with wide coverage. Tweak the router firmware.
Remember, the simplest definition of addiction is a repeat behavior that has adverse consequences. This includes internet.
This might be a good opportunity to discuss why your children have a need to use the internet beyond your parental wishes.