A GaN based LED will produce energies around 3.4eV, depending on Indium and Aluminum content. Higher Al content will produce shorter wavelengths and thus higher photon energies. Indium content reduces the wavelength, but improves efficiency. GaN LEDs are available over a large range of wavelengths and likely are suitable for your application.
Doesn't it also depend on the quantity of photons emited by the led? How many photons does the led emit and what is the quantum yield of the reaction (mol Product/Einstein)? I used a simple UV lamp to remove a OH- from malachite green alcohol form but it could be not enought, or too much, for 2-nitrobenzyl. Do a quick research on "quantum yield deprotonation 2-nitrobenzyl".
Technically it depends on how many photons are at 360nm and at shorter wavelengths. Typically there is a tight distribution of photons around the band-gap of the material, and the higher the energy, the tighter that distribution will be. However, most chemical processes respond to the wavelength in question and shorter wavelengths. As long as there is an ability to conservation of momentum. This implies that for photon energies above the requirement of 360nm (ie short than 360nm wavelengths) phonons need to be part of the reaction to balance the momentum. This becomes a 3 particle problem that might reduce reaction rates if the wavelength is too far off (much off from KbT energy). The intensity of the LED will only affect the reaction rate, not if the reaction will occur (basic quantum mechanics). If intensity is an issue, one can use more LED's, optics that concentrate the light, and smaller volumes of solutions to address the issue.
From the LED-technology side there is a sharp step in output power at 365nm. LEDs with a shorter wavelength have a much much lower optical output power than LEDs with longer wavelengths. That is due to the transition from InGaN to AlGaN. So be sure that you really get 360nm and lower from your light source. The rest is already written.
Did you measure the UV/vis epctra of your compound? If so and there is an overlap between the absorption of the compound and emission of the source (here LED) than you should try it. I am sure there is however tons of literature showing that 2-nitroaromatic compounds are indeed doing some photochemistry upon excitation at 366 nm, so 360 nm should work as well. However you should know, that the 2-nitrosoaromatic compounds, the primary product under the conditions are absorbing even more efficiently and soon you will encounter the internal effect filter (i.e. the (side-)product competition for the light). The amount of photons and concentration of the starting material (i.e. the absorbance of the sample) will both affect the apparent rate of the reaction, but has nothing to do with the quantum yield or how long you need to wait for reasonable conversion. Quantum yields for degradation of 2-nitroaromatic compounds are typically in the range 0.1-0.9, depending on many things, and the time-course of your particular reaction should be followed somehow (tlc, gc, hplc, nmr or uv/vis spectrometry) to give you an idea what's happening. Do not leave the irradiation overnight or so (even if it suggested by some publication you are following, because surely your sample, your source of light, your path of light and many other condition will be different from that reference), unless you have confidence in what you are doing, because the 2-nitrosoaromatic compounds are very reactive (both thermally and photochemically) and could easily deteriorate your product in some secondary reaction(s). A good portion of scepticism in the beginning will be your best freind in pushing the reaction towards success (and that I am saying as a photochemist). Good luck!