jmc

The intersection of the asymptotes is $(5,7),$ so this is the center of the hyperbola. Since the slopes of the asymptotes are $\pm 2,$ the equation of the hyperbola can be written in the form $$(x-5{)}^2-\frac{(y-7{)}^2}{4}=d$$for some constant $d.$ Setting $x=4$ and $y=7,$ we get $d=1,$ so the equation is $$\frac{(x-5{)}^2}{1}-\frac{(y-7{)}^2}{4}=1.$$Then $a^2=1$ and $b^2=4,$ so $c^2=a^2+b^2=5,$ which means $c=\sqrt{5}.$ Therefore, the distance between the foci is $2c=\boxed{2\sqrt{5}}.$