If I had to guess what a dictionary of ideas depart from, I'd expect to see
empeñar = poner de uno mismo un honesto esfuerzo para llevar a cabo una tarea o alcanzar un objetivo
Several new meanings or uses derive from that
empeñar la palabra = comprometerse mediante el acto de dar una promesa (to pledge one's world)
empeñar un objeto de valor = obtener dinero en préstamo dejando el objeto en garantía (to pawn something)
The last action can be reverted, so we use the prefix des-
desempeñar un objeto de valor = recuperar el objeto mediante la devolución del dinero prestado más los intereses acordados y dentro del plazo estipulado (to redeem the object)
But you can't use the prefix des- in this sense with any verb that implies an irreversible or irreparable action. Kids talk about desromper* something because they lack the proper words (reparar, componer, arreglar).
Empeñar is one of such verbs, so, what does des- do with it? In my understanding des- is there to point out the stream of empeño that is poured into some task or office "with all the responsabilities and privileges attached to it", so
desempeñar = (perform, carry out, execute)
I'm sure you will be able to sort out the many meanings of both verbs following these criteria.
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