I have a mid 2011 MBA too and since the update - I get nothing. Downloaded the update, was installing and went to restart and didn't. Since then I've tried to restart it a couple of times, sounds like it it tries to start but after a few seconds - silence. Have left my charger at work so hoping it will solve it in the morning.
Updated: Resetting the SMC worked for me too and the firmware update is showing as installed. Worrying experience though.