How to tell which iPhone app eats bandwidth? Sometimes I notice that the Internet connection in my house becomes REALLY slow.

Disabling WiFi on my wife's iPhone fixes this.

But how can I tell which app/process does this?

iCloud backup is disabled. Dropbox is not installed. The "data usage" report shows cellular data, not wifi, I assume. What do I do? How do I debug this?
If your router doesn't have a way to log bandwidth usage, your best bet would be to run every suspect app on its own.

By that, I mean you should go into the App Switcher (double-press the Home button) and swipe upwards on all of the apps in that list. That kills all of the apps, so the only ones that'll be running are the ones you open afterwards. Kill all of the apps, then let one run and see if it has any effect on your wifi.

Not a great solution, I know, but to my knowledge, there isn't an easy way to figure out your problem.