This is a nice problem i came across yesterday and an attempt of a solution.
Let $X$ be a Banach space with $X=\bigcup_{n=1}^{\infty}A_n$ and each $A_n$ weakly compact then $X$ is reflexive.
Ok so if each $A_n$ is w-compact this means it is norm closed and bounded (Principle of uniform Boundeness).So the complete space $X$ is a union of closed sets hence from Baires theorem one(let $A_L$) has nonempty interior .
Does this mean that it contains the unit ball $(B_X,w)$ which is compact as closed subset of w compact set?
Your argument is almost correct. We just do not know that your set $A_L$ contains the unit ball $B_X$. However, we do know that it must contain some closed ball $B(a,r)$, for some $a\in A_L$ and $r>0$. However, scaling and translation are homeomorphisms, which will give you what you need.