Robber steals insomniac.. does the new insomniac wake up during the night when insomniac is called or do they not act a second time?
You always and only do the action of the card you went asleep as. (With few exceptions.)
The Robber won't wake up during the Insomniac night action because they already did their action. The Insomniac will wake up and see their card - which will most likely be the Robber's.