Standby Instruments powered from the standby bus, powered by the emergency sources - in order of preference - the APU, The HMG which is itself powered by the RAT that provides centre system hydraulics, (without which the flight controls would be useless and you're all gonna die) and finally, all else failing (i.e. the APU won't start and the HMG fails even though there is hydraulic pressure) the battery powered static inverter. The standby magnetic compass, with its pitiful accuracy of +/-5° is enough for an emergency diversion.
As to regulators grounding an aircraft for not having a valid deviation card, too right; but we do have them. As I said we perform a swing and issue a card upon fitting a new standby magnetic compass (its the same as the ones you can buy for a car, and only a little better for accuracy) and it gets an annual check air-swing. The crews fill in the data on the air swing record sheet and we enter the data on a swing sheet, do the arithmetic to extract the A, B and C figures and re-validate the correction card.