Set $V:=L^2$ and $H:=W^{-1,2}$ and define the *maximal regularity space*
$$MR_2(0,T;H,D):=W^{1,2}(0,T;H)\cap L^2(0,T;D),$$
where $D$ is the domain of the unbounded operator on $H$ associated with the quadratic form $Q(u):=\|u\|_V^2$. Then it is well-known (reference in some book by J. Lions, but now I cannot find an exact one) that $MR_2(0,T;H,D)$ is continuously embedded in $C([0,T];V)$. This is *almost* what you want. Not quite (I have not checked, but in this case it looks like $D=W^{1,2}_0$, unlike in your assumptions; and, worse, you are not assuming $u$ to be in $W^{1,2}(0,T;H)$), but also not far away, if you are able to use your assumption that $u\in L^\infty$ to define an equivalent norm on the set of those
$$\{u\in L^2(0,T;W^{1,2}) \hbox{ s.t. } u_t\in L^2(0,T;W^{−1,2})}.$$