You can digitize the border of the polygon. Just make sure of
a) Set the proper snapping options at settings/snapping options
for this case I will snap on the vertex of the existing polygons with a tolerance of 15 pixels. More important, I set that I do not want intersection of new polygons (avoid intersections)
Once you have set that, select your layer, put it in edition mode. The following images are an example
This is what you have
Now I digitize intersections on every boundary of the existing polygons
Notice that I invaded the existing polygons, but I define an intersection for every common border. Also notice that the island is completely coverd with the new polygon (the red one)
Since you defined that you did not want intersections of new polygons, these are erased, keeping perfect common borders and also your island.Save your editions and that should do it.
In this image I erased the island to show that the polygon that I digitized preserved the Island as a single feature. Notice that the digitized polygon contains the island "hole".
hope it helps