LM7812: 15.4V
LM7805: 1.4V
The input for the 7812 looks right, but the input for the 7805 isn't right. Loosing the 5V supply would knock out the display and all, I'd remove the 7805 from the circuit board and then see if you have a better input voltage with the 7805 out of the circuit, probably 9-15V. Most likely the same input as the 7812, 15V. If you do, the 7805 needs to be replaced and you may be lucky.
If the input voltage for the 7805 is still low after you remove the 7805, then you'll need to trace back and find out where it's losing it at. But first, remove the 7805 from the circuit and see if your input voltage comes back up. You can check the 7805 out of circuit with your meter too, it should read as a diode pretty much, from either outside leg to the ground. If it reads shorted from either of those legs to the ground pin, it's bad.
Probably the same deal with the 7812 too, seeing as how the output is only 2.26 volts. Pull them both out of circuit and see if your input voltages look right then. If not, then something between the transformer and the regulators is messed up, you have the 15VAC supply from the trans, so it would be something in between, if it's not the regulators pulling it down. I'd guess it's only the regulators, seeing as how the input of the 7812 looks right, but it may have also gone beyond them.
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You have lightning like we have here! We've had it spark on our window screens, a ball of lightning blow out of a old phone receptacle and fly down to a wall plug, nasty stuff. When you feel that charge in the air, watch out!
Another thing to check, is anything in the circuit, especially the regulators, really hot to the touch?