jmc

The sum of the distances from $(4,-2)$ to the foci of the ellipse is $$\sqrt{(4+1{)}^2+(-1+2{)}^2}+\sqrt{(4+1{)}^2+(-3+2{)}^2}=2\sqrt{26}.$$This is also equal to the length of the major axis of the ellipse. Since the distance between the foci is $2,$ it follows that the length of the minor axis of the ellipse is $\sqrt{(2\sqrt{26}{)}^2-2^2}=10.$

The center of the ellipse is the midpoint of the segment containing points $(-1,-1)$ and $(-1,-3),$ which is $(-1,-2).$ Since the two foci have the same $x$-coordinate, the vertical axis is the major axis. Putting all this together, we get that the equation of the ellipse is $$\frac{(x+1{)}^2}{5^2}+\frac{(y+2{)}^2}{(\sqrt{26}{)}^2}=1.$$Thus, $a+k=5+(-2)=\boxed{3}.$