How can I apply Planck's law to calculate the photon energy density of a warm room?

I am trying to calculate the rate at which photons of any particular frequency will pass through a volume in a room illuminated by black body radiation only.

I've found a couple of starting points but now I'm stuck putting it together.

I could start with $B_\lambda(\lambda, T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$ and pretend all the radiation is coming from one spot on the wall, then multiply it by the effective solid angle to the volume and the volume itself, but I'm not sure if this is reasonable.

Also since I'm counting photons it seems I want to take a definite integral over the relevant frequency range. In which case, setting $x=\lambda$, $k=2kc^2$, and $q=hc/(k_BT)$ I'll end up with this less-than-elegant answer:

I suspect there's an easier way. Is there a simple formula for the energy density of empty space when the volume under examination is "far" from the walls?

To restate the problem: take a small volume of space within a room whose walls are emitting black body radiation at a known temperature; the size and shape of the room is known (we can assume any shape which is convenient, if that matters) -- how many photons per second with wavelength $\lambda_{min}<\lambda<\lambda_{max}$ are passing through this volume?