Deriving a simpler expression for $\left({I}-T\otimes T\right)^{-1}\left(T\otimes T\right)$ using Kronecker product properties
Solution 1:
In general, if $M$ is invertible:
$$(I-M)^{-1}M=(M(M^{-1}-I))^{-1}M=(M^{-1}-I)^{-1}M^{-1}M=(M^{-1}-I)^{-1}$$
giving here the alternative form, in the case $T$ (therefore $T \otimes T$) is invertible:
$$\left(\left(T\otimes T\right)^{-1} -I_{n^{2}}\right)^{-1}$$
with $\left(T\otimes T\right)^{-1}=T^{-1} \otimes T^{-1}.$
Is it really "simpler" ? It certainly depends on the use you will have of this expression.