The correct way to build this driver is, with a working internet connection:
sudo apt-get install build-essential linux-headers-generic
cd ~/Desktop/AX88179_178A_LINUX_DRIVER_v1.9.0_SOURCE
...or wherever you extracted the package, if not desktop.
make
The driver ax88179_178a is present in Ubuntu 13.10, so if you re-install, compiling will be unnecessary.
Most Makefiles have a process to make, clean, install and uninstall. Yours evidently does not. First, let's test the driver to see if it drives the device correctly:
cd ~/Desktop/AX88179_178A_LINUX_DRIVER_v1.9.0_SOURCE
sudo modprobe usbnet
sudo insmod ax88179_178a.ko
Does your device work correctly? If so, we'll install it the hard way:
sudo cp ax88179_178a.ko /lib/modules/`uname -r`/kernel/drivers/net/usb/
Those backticks are on the left side of my US keyboard on the same key with ~. This will overwrite your previous attempt, if any.
It seems that your device connects and immediately drops. I noticed this bug report: https://bugs.launchpad.net/ubuntu/+source/linux/+bug/1269883 Evidently, the poster had better luck with a USB 2.0 port. Please try.
Aside from that, I suggest you contact ASIX, as they are the author of the driver we compiled.