Due to a theorem of Nakayama, a finite dimensional algebra $A$ over a field $\mathbb{k}$ is self-injective provided that each simple right module of $A$ is the socle of a projective indecomposable right module of $A$ (the converse holds as well). The only proof I am aware of argues with Morita Theory. I wonder if there is a more direct approach.
Here is a sketch of the proof in the book of Skowroński and Yamagata:
Let $e_1, \dots, e_n \in A$ be a complete set of mutually orthogonal and non-isomorphic primitive idempotents. Each simple right module of $A$ occurs exactly once as the top of some $e_iA$, and by the assumption above, each occurs exactly once as the socle of some $e_jA$. Hence, there is a “Nakayama permutation” $\pi \in S_n$ such that $\mathrm{top}(e_{\pi(i)}A) \cong \mathrm{soc}(e_iA)$. Now $e_iA$ embeds into the injective hull of $\mathrm{soc}(e_iA)$, which is $D(Ae_{\pi(i)})$, where $D = \mathrm{Hom}_\mathbb{k}(-,\mathbb{k})$ is the usual duality. By setting $e = e_1 + \dots + e_n$, we get a monomorphism $\varphi \colon eA \hookrightarrow D(Ae)$.
At this point the authors argue that $\varphi$ induces a monomorphism $eAe \hookrightarrow D(eAe)$ of right $eAe$-modules which must be an isomorphism by comparing dimensions. So $eAe$ is a Frobenius algebra, and $A$ must be self-injective being Morita equivalent to $eAe$.
Can we come to the conclusion more directly by showing that $\varphi \colon eA \hookrightarrow D(Ae)$ must already be an isomorphism? Of course it suffices to show that $eA$ (the minimal projective generator of right $A$-modules) and $Ae$ (the minimal projective generator of left $A$-modules) have the same dimension.