You realize, don't you, that because both lines have direction vector <1, 1, 3?> the two lines are parallel?
Take any point on the first line, say take (3, 1, -2) with m= 0. The plane perpendicular to the first line, through that point, is 1(x- 3)+ 1(y- 1)+ 3(z+2)= x+ y+ 3z+ 2= 0. The second line, x= 1+ n, y= n, z= 1+ 3n, intersects the plane where 1+ n+ n+ 3(1+ 3n)+ 2= 11n+ 6= 0 so n= -6/11. So the point is x= 1- 6/11= 5/11, y= -6/11, z= 1- 18/11= -7/11. The distance between the two lines is the distance between the points (3, 1, -2) and (5/11, -6/11, -7/11).