Try boosting it from another (running) car. My bet is that there isn't enough cold cranking amps available from the battery to turn the starter/flywheel.
Whenever a battery is basically dead, I've experienced a clicking under the dask area, often with a working radio/lights, but NO sound from the starter. Swap the battery and voila she fires up.
Attempting to start it from a booster cable connection should rule this out. Let it sit, connected to the running vehicle, for about 3-4 minutes, then try turning it over.