What is the prognosis for a brain stem stroke, 7 days ago, still on ventilator?

It depends.: The prognosis depends on many variables, such as the patient's age, other health conditions, and underlying cause of the stroke. However the need for mechanical ventilation a week out is, in general, not a good sign.

Indefinite: In general coma from a brainstem lesion lasting one week bodes poorly. If diaphragmatic weakness, without coma is present, the prognosis is only slightly better.