Connecting a Nexus 7 (2013) to a VGA projector
If you have a laptop with USB ports that can also connect to the VGA projector, then you can use Droid@Screen (http://droid-at-screen.ribomation.com) to show the Nexus' display on the laptop and in turn, on the projector. 

Depending on what you are doing, this might work while you figure out a more robust solution. I say this because the main downside is lag, so it's not a good solution if you are trying to project video or show off a new buttery interface.