I don't have any direct experience with this model but I can throw out a few general things.
Current draw is in the normal range, so nothing enlightening there.
If the processor is alive, you'll find some slow (50 - 150 Hz) square waves or stair-step signals on all of the LCD lines. Doesn't sound like it's alive in this case.
Examine the board carefully with light and magnification. Never know what you may find. Bad solder joints and spots of copper erosion are possible faults. Don't forget to examine the switch rotor and contacts too.
Try to find data sheets for all of the little off-the-shelf chips on the board. The function of these chips may not be obvious, but at least you can find the power/gnd pins. You will probably find one or more voltage regulators, providing a place to check power rails. Use the COM jack as reference for analog ground. The analog measurement section usually has a bi-polar supply. The processor will probably have a +3.3V supply.