general relativity - Hulse-Taylor binary pulsar - what is the rate of mass/energy loss from the source?

For reference, the relevant bit from the paper is:

The observable pulsar is a weak radio source with a flux density of about $1,mathrm{mJy}$ at $1400,mathrm{MHz}$. ... Our most recent data have been gathered with the Wideband Arecibo Pulsar Processors (“WAPPs”), which for
PSR B1913+16 achieve $13,mathrm{mu s}$ time-of-arrival measurements in each of four $100,mathrm{MHz}$ bands, using $5$-minute integrations.

Those are millijanskys, aka milli flux units, so that the flux density is about
$$1,mathrm{mJy} = 10^{-29},frac{mathrm{W}}{mathrm{m}^2cdotmathrm{Hz}}text{,}$$
and hence the detected irradiance is on the order of $10^{-27},mathrm{W}/mathrm{m}^2$. Since we're about to make rather uncertain assumptions anyway, I won't bother worrying about doing more than an order-of-magnitude calculation.

Assuming that this flux is uniform across a conical beam with cross-sectional radius 5 arc degrees and assuming that the source is 21,000 light years from Earth ... - How much energy is emitted (per second) from the source in the beam?

A spherical cap has surface area of $A = 2pi Rh$, and here $R = 21,mathrm{kly}$. Now, I'm unclear what cross-sectional radius means if measured as an angle, but I take it to mean that the opening half-angle of the cone is $frac{vartheta}{2} = 5^circ$, in which case
$$A = 2pi R^2left(1-cosfrac{vartheta}{2}right) sim 10^{39},mathrm{m}^2text{.}$$
Thus, the power would be $Psim 10^{12},mathrm{W}$, but note that in addition to the assumptions you've just listed, we're only talking about a particular radio band.