I was playing Guns of Icarus and followed a route all the way to the "Into The Breach" level. But that level seems to be a never-ending survival one.
Is there any way to actually finish the game? Is there any ending of some sort?
Perhaps unfortunately, there is not an ending to the story. I have played the game, and Into the Breach is meant to be the final apocalyptic mission. It's your suicide mission into the pirates, and does not end until you die; you just try to accumulate a high score.
If you're looking for references, this review calls Into the Breach "survival mode" which is just what I have described above. This survival mission is also the end goal of the game. Everything else is just a build up to upgrade your ship in preparation.