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.