How many photons are emitted by a lightning strike?

From How Big Is A Lightning Bolt? we see that a lightining bolt is “an inch wide and five miles long”, and at “50,000 °F”. So in useful units, approximately 3 cm diameter, 8 kilometer long, 28000 K hot.

If we consider that the heat is mostly due to black body radiation (for a perfect black body with an emissivity of $\epsilon = 1$), then the power will be given by the Stef-Boltzmann law:

$$P = A \epsilon \sigma T^4$$

The area, $A$ of the lightning bolt (a cylinder, of course) is given by

$$A= 2 \pi\times(3 \text{ cm})\times 8 \text{ km} \sim 1500 \text{ m}^2$$

And so,

$$5.2 \times 10^{13}\; \text { Watts of power.}$$

Lets say, it lasts 10 miliseconds, so its around $\sim 5 \times10^{11}$ J.

Now to calculate it the amount of photons properly, you would have to consider the spectrum of the black body radiation, and convert the energy density to number of photons using Planks law. I will just use the rule of thumb that "1 Watt of monochromatic visible light is approx $10^{18}$ photons per second".

And so, it would be around:

$$\sim 10^{29}\ \text{ photons.}$$

According to Could We Harness Lightning as an Energy Source?:

An average bolt of lightning, striking from cloud to ground, contains roughly one billion ($1,000,000,000$) joules of energy.

According to Visible light:

Red photons of light carry about $1.8$ electron volts (eV) of energy, while each blue photon transmits about $3.1$ eV.

So let's take an average photon energy of $2.5 \text{ eV}$.
Assuming all the energy of the lightning is converted to visible light, we can calculate the number of photons.

$$ N = \frac{10^9 \text{ Joule}}{2.5 \text{ eV}} = \frac{10^9 \text{ Joule}}{2.5 \cdot 1.6 \cdot 10^{-19} \text{ Joule}} = 2.5 \cdot 10^{27}$$

This looks like it's a quantity that we don't have a particularly good grip on. Quoting from Rakov & Uman's Lightning: Physics and Effects (Cambridge University Press, 2003),

An approximate range for the electrostatic energy available for a lightning flash lowering a charge Q to ground can be evaluated by multiplying Q by the upper and lower limits for V, the magnitude of the potential difference between the lower boundary of the cloud charge source and ground. Assuming that Q = 20 C, thought to be typical for a cloud-to-ground flash, and using the range of V from 50 to 500 MV estimated earlier in this section, we find that each flash dissipates an energy of roughly 1 to 10 GJ (gigajoules). Note that a flash is typically composed of three to five strokes, and that the first stroke is usually a factor 2 to 3 larger (in terms of peak current and peak field) than a subsequent stroke, that is, any stroke other than the first. The above energy range inferred from electrostatic considerations is for all processes involved in a lightning discharge. Specifically, this energy estimate may well be dominated by the energy dissipated in the formation of numerous filamentary channels in the cloud that serve, in effect, to funnel cloud charges into the narrow channel to ground. Marshall and Stolzenburg (2001), from their balloon soundings of the electric field through thunderstorms and assumed minimum and maximum values of charge transfer, estimated the energy available for lightning to be in the range from 10 MJ to 10 GJ, the energy available for intracloud flashes (Chapter 9) being usually larger than that available for ground flashes. There is no consensus regarding the proportion in which the total return stroke energy is converted to thunder, hot air, light, and radio waves. According to Paxton et al. (1986), who used a gas dynamic model of the lightning return stroke (subsection 12.2.2), almost 70 percent of the total energy input to the channel is optically radiated from the channel. However, Few (1995), in his theory of thunder (subsection 11.3.2), assumes that essentially all the input energy is delivered to a shock wave 116 4. Downward negative lightning discharges to ground that subsequently is heard as thunder. As discussed in the first part of subsection 12.2.6, the total lightning energy input estimates of Paxton et al. (1986) and others, who employed gas dynamic models, differ from that of Few (1995) by two orders of magnitude or so.

Krider and Guo (1983) and Krider (1992) estimated that the radio-frequency power radiated by a subsequent return stroke at the time of the field peak, 3 to 5 GW, is about two orders of magnitude greater than the optical power radiated in the 0.4 to 1.1 µm range at the time of the field peak. The average zero-to-peak risetime of the subsequent stroke field waveforms was 2.8 µs. The total optical power, however, was found to dominate at later times, the peak optical power occurring about 60 µs after the electric field peak (because the risetime of the optical signal was determined by the geometrical growth of the return-stroke channel)

(emphasis added).

That said, it does look like the estimates of about 100 MJ to 10 GJ radiated as optical power capture the rough ballpark; assuming a photon energy of 2.5 eV gives a rough total of some $10^{25}$ to $10^{28}$ photons per lightning strike as a starting ballpark estimate.