How to explain the discrepancy between the calculated absorption wavelength within the particle in the box theory and the observed wavelength?
Luckily I found a source containing your molecule as an example.

You need your equation for the energy difference between the HOMO and the LUMO, which is (nearly) defined as you wrote:
$$\Delta E = \frac{(N+1)h^2}{8 m_e L^2}$$

As you want a wavelength, you need to convert the energy difference, which yields then:
$$\lambda = \frac{8m_ecL^2}{h(N+1)}$$

Now they state, that the length of the box can be guessed by the amount of carbons (p) between both nitrogens by using this formula:
$$L = (p+3+\alpha)l$$
... with $l$ being the length of typical benzene-C-C-bonds, thus being 0.139 nm and $\alpha$ is an empirical correction factor for polarizable end groups.

The amount of pi-electrons (N) that you need to inlude can be calculated by a similar equation:
$$N = p+3$$
... where the additional "+3" come from both nitrogens (would be actually +4, but it's +3 because one nitrogen atom had to form an ion).

This leads to their equation 4:
$$\lambda = 63.7\frac{(p+3+\alpha)^2}{p+4}$$
... where all the physical constants have been combined to give 63.7.

As you have three carbon atoms between both nitrogens, you come up with 328 nm and when you include their "correction factor for polarizable end groups" of $\alpha = 0.675$, you come up with 405 nm, which is more than close to the experimental result.



When I use your box length and use 12 electrons (3 from the "left" benzene ring, 4 from the two double bonds to the "right" nitrogen" and 1 from the lone pair of this nitrogen), I come up with about 356 nm, which is not that far away as it sounds. In eV this is an error of 20% (the 405 nm are wrong by about 5%), whereas your 200 nm would have been an error of 120%!