Using Solve[] to find Eigenstates of a 1D Double Dirac Potential

I’d like to Solve

$$ k^2 equiv – frac{2mE}{hbar^2} = (- frac{mA}{hbar^2} (1+ e^{-2ka}))^2 $$

for E, in terms of m, $hbar$, A, a.

I tried using the following command:

Solve[-((2 m ene)/h^2) == (m^2 A^2)/h^4 (1 + E^(-2 a*Sqrt[-((2 m ene)/h^2)])), ene]

Isn’t working well for this task. What do you recommend? At first glance it seems it could not be simple to solve "by hand".

Background: This problem comes from Solving a 1D Quantum well with 2 Symmetric Dirac’ Deltas $delta_a$ and $delta_{-a}$, where $A$ is the amplitude.