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?
Yeah, I had a similar problem 2 boards ago where I had a ground plane on one side, but had to run large traces between some sections of it to connect it all together. The last board I just hand routed since I don't like the Eagle auto-router (Or I just don't understand how to properly set it up). Diptrace has a surprisingly good one though. I do usually end up going through and rearranging wires afterwards, making them all one plane to reduce vias, and such.
I would suggest running a rather wide trace between the two pins, at the very least, to connect the ground plane sections. Or run it to two vias at the edges between them.