Can split-brain patients sleep like whales? Regular split-brain patients still have some remaining connection between the hemispheres, but would it be possible for their hemispheres to fall asleep independent of each other?

What about when not just the corpus callosum, but also the anterior and posterior commissures are not functioning for interhemispheric communication?
Short answer
An animal study did not reveal any effects on the synchronization of the cerebral hemispheres after callosotomy, as based on EEG recordings. When one hemisphere showed sleep patterns on the EEG, the other half would show similar activity.  

Background
If intact commissures are needed between the two hemispheres to synchronize the two halves of the brain, a callosotomy would be expected to result in different EEG patterns during sleep. The most extreme case being laid out in the question, namely one hemisphere being in a different arousal state than the other.

In a study in split-brain cats (Susic &amp; Kovaevic, 1974) the authors performed a callosotomy or related surgeries, including the sectioning of the anterior and posterior commissures*, and recorded the EEG during various behavioral states, including wake and sleep states and their transitions. After separation of the hemispheres the authors did not find any differences in EEG patterns across the two hemispheres in cats. Specifically, during transition from the alert to the drowsy state, the onset of EEG slowing appeared simultaneously in both hemispheres in all cats regardless of the type of section made. The shift from the synchronized to desynchronized pattern occurring during arousal or at the beginning of the paradoxal sleep (REM sleep) was always simultaneous in both cerebral hemispheres. The transition from REM to short arousal or to slow-wave sleep patterns was also bilaterally symmetrical. These results demonstrate that shifts in behavioral states from vigilance to sleep and vice versa are characterized by a simultaneous bilateral symmetry of the EEG events. 

* List of surgical groups (n= 14 per group)  


Cats with sections of commissural systems between hemispheres and diencephalon (corpus callosum, fornix, commissura hippocampi,
septum, commissura anterior, massa intermedia of thalami, commissura posterior, commissura habenularum, commissura intercollicularis).
Cats with interrupted left optic tract and sections of the commissural system between hemispheres and diencephalon.
Cats with an interrupted left optic tract and section of the left half of the midbrain tegmentum (the lesion passed intercollicularly in the
dorsal part, and ventrally into the premesencephalic area).
Cats with an interrupted left optic tract section of the left half of the midbrain tegmentum and section of commissural systems, combined with the section of crossed fibers of the commissura supraoptica. 


Reference
- Susic &amp; Kovaevic, Brain Res (1974); 65: 427-41