First I checked the value of

Sqrt[2] Pi^(3/2) / Gamma[1/4]^2

which actually evaluated to:

0.59907

which is not quite what the paper says. I also noticed that the paper is using K and E to denote the complete elliptic integrals of the first and second kind respectively. Unfortunately elliptic integrals can have different notations depending on whose definition is being used. In particular Mathematica (and Abramowitz and Stegun) uses:

$$K (m)=\int _0^{\pi/2}\frac{1}{\sqrt{1-m \sin^2\theta}} d \theta$$

while some others use:

$$K (m)=\int _0^{\pi/2}\frac{1}{\sqrt{1-m^2 \sin^2\theta}} d \theta$$

I assumed that the difference you get is likely due to this, so I tested it (note the squared argument now):

N[Sqrt[2] EllipticE[(1/Sqrt[2])^2] - EllipticK[(1/Sqrt[2])^2]/Sqrt[2]]

which also returns:

0.59907

For clarity of course, I would recommend contacting the authors to determine which definition of the elliptic integrals they used (it would have been nice if they included to reference for it).