This is easily solvable, I've had a few issues like you since the Nexus 5 does not have an SD Card slot either.
What you'll need to do is set up ADB, Google it if you don't have it.
Then go to where you have the contents (adb files). Or you can go into cmd (command promt) and type in "cd your/adb/folder/destination" if you haven't done the "cd" method simply go into the adb folder. And open a cmd window there, once open. Boot your device into recovery mode and plug in your cable to your PC. So that it detects your device.
Then in the command prompt window type the following command - "adb devices" (This will run adb and it'll check to see if adb is properly installed, if your device does not pop up, make sure you cable is in properly. If it still doesn't show, you will have to install the drivers for your device again... Properly).
Now it should list your device, go into the G2 section and download the preferred rom.
Once downloaded simply type this in the command prompt window.
Adb push path/to/rom.zip /sdcard
Eg - adb push c:/user/user/Downloads/cm11.zip /sdcard
Now this should push the .zip to your internal memory, in the recovery mode (after adb has down its job). Simply wipe data and install the .zip c:
Your device should work.
If I helped, don't hesitate to hit that Thanks button