Yes, Ground plane on both layers is best.
Add some vias and name them GND also to connect the layers in places where there are no pins to connect the layers.
There is a via between pad 8 and pad 3 - if you move that to the left of pad 1, then the bottom GND layer will connect pad 8 and pad 1. Cross under the vertical trace, go on the top layer and then back to the bottom layer.
I don't recall the autorouter ever using traces to connect GND pads up; maybe it does that if there are no GND planes?