You could try and use Live paint.
But first make a copy.
Then go to Object > LIve Paint > Gap options and set the gap detection to a high values.
Then take the live paint tool and click inside the gaps. Does that work?
The other option would be to do Object > Path > Offset path.
Offset it so much that all the gaps are closed.
It will make a copy automatically.
The copy is selected afterwards
Then in the pathfinder panel hit Unite (top left)
Then use Offset path again, this time using a negative value.