SG electret microphone static impedance

Here's the circuit you linked to with a couple of red circles on it: -

Check that capacitor C1 isn't somehow leaking some DC current through to the -Vin pin.

Also check that R4 is fitted correctly and you can measure the same voltages on each of it's pins as you do on the op-amp pins. R4 could be broken - can you measure it with a meter?

Measure that copper tracks are fine between R4 and the op-amp.

If necessary lift C1 from the circuit to check that -Vin and +Vin become the same level.

The OPA345 is unquestionable a good device for this application if you don't care too much about noise. The AD8606 does the same but will produce less noise.

I think you may be overcomplicating this a little bit - why don't you just connect the electret microphone (observing polarity) to the microphone wires via a 3.3Kohm resistor and see what happens.

Your circuit won't work as it stands because the signal you are trying to put back onto the line gets fed back to the microphone via the 10k and it'll all end up very low gain in my humble opinion.

I don't think there is a substitute for trying it out and if it's not loud enough, putting a 10uF capacitor across the 3k3 resistor I mentioned and if that doesn't sound like the right volume maybe drop the 3k3 to 1kohm.

As for matching the impedance - this won't make a great deal of difference to the kit that recieves the signal. It's likely got an unbalanced input and is probably not differential so I don't think you need to jump thru hoops on this.

If you still believe the mic on its own won't work, try this: -