5% of what? How do you define death by a mentally ill person? Does that include suicide? Most of the mass shootings we've had, more than likely by mentally ill people, also involved suicide of the shooter. If you count only the mass-shooting-related deaths, that might be 5% of the total.
The paper I linked above says in 2015 there were 22,018 suicides by gun out of a total of 36,252 gun-related deaths in the USA. That's 61% – far more than the 13,463 homicides (37%).
"In 2015, 36,252 people died of a firearm-related injury in the United States, approximately the same number of deaths as occurred in motor vehicle crashes. The same year, more than eighty thousand people were non-fatally injured (CDC 2017). The distribution of firearm deaths in 2015 is typical of the distribution over the past several decades: the majority of firearm deaths were suicides (22,018), followed by homicides (13,463) and then unintentional firearm injuries (fewer than one thousand)."
Clearly, the biggest problem is not mass shootings, but gun-related suicides.
Although I agree that the NRA's mental health argument is BS, a smoke screen meant to distract the opposition, the mental health issue is highly related to gun deaths. I don't know how many people among those 22,018 suicides were mentally ill, but I would guess it would be quite a few, as many as 100%. Where in the constitution does it say the government may not regulate sales and manufacture of the weapon of choice for mentally ill people to commit suicide?