If I where to perform that task, I would set up a regular computer/laptop with audio in, and record your input / output signal through right/left audio channel. Of course via a small transformer to isolate the audio input.
I would have used the software Audacity, where you're able to make a stereo track into two mono tracks and invert one of them - you see where I'm going.
Then move one track forward in time until they cancel each other out (the level should be identical for that to work).
At least it is a cheap way of meassuring delay of audio signals.