vestibular rehabiliation is the way to go - looking for nystagmus is unfortunately not really going to be of benefit because the patient will have nystagmus - a central nystagmus due to the central lesion. You could do a full oculomotor assessment to assess for a peripheral problem also but it should just be a central problem.

Unfortunately the cerebellum/central vestibular issues do not respond well to physiotherapy - so you need to ensure safety, lots of education, and teach the patient how to use other available strategies (somatosensation and visual).

I would also work on lots of core stability to help facilitate better anticipatory postural adjustments and also looking at dissociation of the head on trunk.

Hope it all goes ok.