We present a study of the central 2.3 kpc of NGC 5135, a nearby luminous infrared galaxy (LIRG) with an active galactic nucleus (Lambda GN) and circumnuclear starburst. Our main results are based on intermediate spectral resolution (similar to 3000-4000) near-infrared data taken with the SINFONI integral field spectrograph at the ESO VLT. The ionization of the different phases of the interstellar gas and the complex structures of the star formation have been mapped. Individual regions of interest have been identified and studied in detail. For the first time in this galaxy, we have detected the presence of a high excitation ionization cone centered on the AGN by using the [Si VI]lambda 1.96 mu m line. So far, this structure is the largest reported in the literature for this coronal line, extending (in projection) as far as similar to 600 pc from the galaxy nucleus. In a complex spatial distribution, a variety of mechanisms are driving the gas ionization, including supernova remnant (SNR) shocks, young stars and AGN photoionization. The excitation of the molecular gas, however, is mainly produced by X-rays and SNR shocks. UV mechanisms like fluorescence represent a marginal overall contribution to this process, contrary to the expectations we might have for a galaxy with a recent and strong star formation. Our supernova (SN) rate estimations from [Fe II]lambda 1.64 mu m are in excellent agreement with 6 cm radio emission predictions. Typical SN rates between 0.01 and 0.04 yr(-1) were found for individual approximate to 200 pc-scale regions, with an overall SN rate of 0.4-0.5 yr(-1). Even though NGC 5135 has suffered a recent starburst (6-7 Myr ago), the data strongly suggest the presence of a second, older stellar population dominated by red giant/supergiant stars. However, simple stellar population models cannot sharply discriminate between the different populations.