Bandwidth of an energy signal
I'm using the following assumption: the magnitude of the spectrum is given by the formula

$$|X(f)|=\frac{1}{\sqrt{1+(f/B)^2}}$$

Then the energy spectral density is

$$|X(f)|^2=\frac{1}{1+(f/B)^2}\tag{1}$$

The 3 dB bandwidth is simply the frequency for which (1) becomes $0.5$, which is $W_{3dB}=B$. The bandwidth according to the 90% energy/power definition is

$$\int_{-W}^W|X(f)|^2df=0.9\cdot\int_{-\infty}^{\infty}|X(f)|^2df\tag{2}$$

Equation (2) is the reason why I assume the definition as given in (1), because otherwise the integral on the right-hand side of (2) does not converge, so the 90% energy/power definition would not make sense. With the definition (1) the integrals in (2) are easily solved using the $\arctan$ function, from which $W=const\cdot B$ follows, where $const=6.31$ (if I'm not mistaken), but I'm sure you can do the calculation yourself.