I try to show that
$$\sum _{k=1}^{\infty } k^{36} \text{sech}(\pi k)=\frac{41222060339517702122347079671259045}{137438953472}+\frac{i \left(\psi _{e^{\pi }}^{(36)}\left(1-\frac{i}{2}\right)-\psi _{e^{\pi }}^{(36)}\left(1+\frac{i}{2}\right)\right)}{68719476736 \pi ^{37}}$$ using Elliptic theory it is possible to get similar result
This can also be done via elliptic integrals and it is possible to get a closed form involving $\Gamma(1/4)$ but the calculations are complicated. We need to use the power series for $\text{dn}(u, k)$ and compare it with its Fourier series. Thus we have $$\text{dn}(u, k) = 1 - k^{2}\frac{u^{2}}{2!} + k^{2}(4 + k^{2})\frac{u^{4}}{4!} - \cdots\tag{1}$$ and the Fourier series is given by $$\text{du}(u, k) = \frac{\pi}{2K} + \frac{2\pi}{K}\sum_{n = 1}^{\infty}\frac{q^{n}}{1 + q^{2n}}\cos (n\pi u/K)\tag{2}$$ and $q = e^{-\pi K'/K}$. If we put $K'/K = 1$ so that $k = 1/\sqrt{2}$ and $q = e^{-\pi}$ then $$\frac{q^{n}}{1 + q^{2n}} = \text{sech}(n\pi)$$ and in $(2)$ we need to express $\cos(n\pi u/K)$ in power series of $u$ and then equate the coefficients of $u^{36}$ in $(1)$ and $(2)$ to get the value of $\sum_{n = 1}^{\infty}n^{36}\text{sech}(n\pi)$ in terms of $k = 1/\sqrt{2}$ and $K = \Gamma^{2}(1/4)/(4\sqrt{\pi})$. It is guaranteed that the desired sum will be a multiple of $(2K/\pi)^{37}$ and a polynomial in $k^{2} = 1/2$ so that it is a rational multiple of $(2K/\pi)^{37}$. Thus using value of $K$ in terms of $\Gamma$ function we get $$\sum_{n = 1}^{\infty}n^{36}\text{sech}(n\pi) = \frac{\Gamma^{74}(1/4)}{2^{37}\pi^{55}\sqrt{\pi}}\times\text{ a rational number}\tag{3}$$ Note that the number $137438953472$ appearing in your formula is $2^{37}$. Unfortunately finding the "rational number" mentioned above requires us to calculate the coefficient of $u^{36}$ in Taylor series $(1)$ and this is a big big challenge to do by hand. Someone should help us out with Maple/Mathematica.
Update: It appears that there are recursive formulas to obtain the coefficient of $u^{n}$ in the Taylor series for elliptic functions. See the paper On Power Series for Elliptic Functions by E.T. Bell for more details.