I guess you don't need approximate identity assumption to prove it (in any case what does approximate identity mean for a TRO?). I am not sure which paper of Kirchberg you are referring to, but indeed Kirchberg has proved that the sum of a closed left ideal and a closed right ideal is closed (for C-algebras) and this fact generalizes to TROs (C-spaces in Kirchberg's terminology) via the linking algebra construction. For the proof, see Section 4 of Kirchberg's paper "On restricted perturbations..." JFA 1995 (https://mathscinet.ams.org/mathscinet-getitem?mr=1322640).

I guess you don't need approximate identity assumption to prove it (in any case what does approximate identity mean for a TRO?). I am not sure which paper of Kirchberg you are referring to, but indeed Kirchberg has proved that the sum of a closed left ideal and a closed right ideal is closed (for $C^*$-algebras) and this fact generalizes to TROs ($C^*$-spaces in Kirchberg's terminology) via the linking algebra construction. For the proof, see Section 4 of Kirchberg's paper "On restricted perturbations..." JFA 1995 (https://mathscinet.ams.org/mathscinet-getitem?mr=1322640).

Added: Associated with a closed left TRO ideal $I$ is a closed left ideal $$L:=\left[\begin{matrix} [VI^*] & I \\ I^* & [V^*I]\end{matrix}\right]$$ of the linking $C^*$-algebra $$A:=\left[\begin{matrix} [VV^*] & V \\ V^* & [V^*V]\end{matrix}\right].$$ Likewise for a right TRO ideal.