Uh, what does "the number of murders by gun crime" have to do with
"letting the mentally ill own firearms"? Are you suggesting that
committing murder is prima facie evidence that one is mentally ill?
If not then what are you suggesting?

A normal level-headed person doesn't commit murder unless they're sick
- or perhaps to use an old phrase, criminally insane.

Being "mentally ill" can come about in a variety of ways, such as
stress in work / being sacked (going postal), and maybe a little
mentally disturbed, i.e. finding your wife in bed with the neighbour -
having a gun available in these circumstances is likely to lead to a
more permanent ending than, say, a good punch up.

Most gun crime against kids in the UK is not perpetrated by other kids
(although the stats are getting worse) - whereas in the US it's pretty
much endemic.