Commercially available paraffin wax LTP ST, manufactured in Poland by POLWAX, was used as a phase change material (PCM) for passive cooling of an LED panel containing 28 high power light emitting diodes (LEDs). Paraffin wax LTP ST of density ρ = 930 kg·m−3 at room temperature (RT) was chosen over other POLWAX waxes (LUXOLINA, LUXOLINA-ST, and LTP 56-20) because of its melting point range (44.5–55.4 °C), relatively high latent heat of fusion ΔH = 218.8 J·g−1, high specific heat Cp = 2.11 J·g−1K−1 and thermal conductivity k = 0.233 Wm−1K−1 at 0 °C. The thermophysical properties were studied in samples of pure LTP ST paraffin and doped with multi-walled carbon nanotubes (1.99, 3.49, 5.35, and 10.49 wt%, MWCNTs) or silver nanowires (0.26, 0.32, 1.06, 2.10, and 7.35 wt%, SNWs). Analysis of the thermal effects of doped samples showed a relative increase in the degree of subcooling, averaging 100% for MWCNT and 46% for SNW, a relative 15÷25% decrease in enthalpy of melting for MWCNT and 14÷16% for SNW. A 44% increase in thermal conductivity was found for the sample containing 5.35 wt% MWCNTs and a 91% increase for 1.06 wt% SNW. The results of cooling efficiency tests for three types of developed heat sinks fabricated of AW-2017A aluminum alloy are presented, i.e., (a) full system without PCM filling, (b) system with PCM chamber without intracellular ribs, (c) and system with PCM chamber with intracellular fins.