How can we find the simple modules of this algebra
Denoting this algebra as $R$, the simple $R$ modules are exactly the simple $R/J(R)$ modules, where $J(R)$ is the Jacobson radical you have found. This is isomorphic to the product ring $k^3$. Now you just need to conclude what the isoclasses of simple modules over that ring are.
The same strategy was applied at this solution, for example: https://math.stackexchange.com/a/147195/29335
And why this algebra is not semisimple
You just said that the Jacobson radical is some nonzero set, but the Jacobson radical of a semisimple ring is zero.
i,e it is isomorphic to the algebra $k^5$????)
There is more than one semisimple $k$ algebra than $k^5$. It could have been $M_2(k)\times k$ or $Q\times k$ if $k$ admits a $4$-dimensional quaternion $k$-algebra other than $M_2(k)$.