The formation of kinks in a loop of bundled polyelectrolyte filaments is analyzed in terms of the thermal fluctuations of charge density due to polyvalent counterions adsorbed on the polyelectrolyte filaments. It is found that the counterion-mediated attraction energy of filaments depends on their bending. By consideration of curvature elasticity energy and counterion-mediated attraction between polyelectrolyte filaments, the characteristic width of the kink and the number of kinks per loop is found to be in reasonable agreement with existing experimental data for rings of bundled actin filaments.