Hi
Emptying a car battery in just 2 hours requires a current of approx. 30 amps. Surely this current has to run throug some system(s) in the car. Leaving the ignitinon on will activate display, BII-unit, airbag system, engine control system(?), interior lights(if switched on), fuel pump, radio-unit, ac-control unit and defroster fans.
Will these systems consume 30 amps alltogether? I don't know for sure, but its possible.
Regardless of this, you might have some other problem, and yes it is possible that the battery voltage is to low for the display to "start". One shorted battery-cell will give a batteryvoltage of 10volts, which could be enough to start the car, but might cause electronics to misbehave.
You should start by measuring the batteryvoltage:
Ignition off:-should measure no less than 12 volts.
If less than 12 volts:
-either the battery is defective(1 shorted battery cell?)
-or there is some shortcircuit in the car(pre ignition relay).
To check battery, disconnect terminal(s) and measure voltage. If less than 12 volts, recharge and measure again. Still below 12volts? -buy new!
-To check for shortcircuits, (battery is ok, but voltage drops when battery connected to car) - check current consumption in the car. This requires a good amperemeter, capable of handling both low(milliamps.) and high(10 -25 amperes)currents(or lots of spare fuses for amp.meter....).
When ignition is turned off, no more than 100milli-amperes shoult be read.
Ignition on:-Battery voltage should keep above 11,5volts.
-Lower voltage indicates shortcircuits.
Engine running:
-Battery voltage should be between 13,5 and 14,5 volts.
-below 13 volts indicates poor charging(generator) or very high current consumption(shortcircuit).
-display will not function properly below 10,5 volts
Locating short-circuits with ignition on, requires connecting high-amp amp-meter. Then remove fuses in car until amp-measurements drop. All fuses removed and still high current = faulty wiring-harness(deep shit
)
Hope this can be of some help.
BTW: Apparently drained battery(car seems dead), could simply be a loose or corroded batteryclamp. Just connecting/fiddling with the batterycharger could be enough to restore connection.