You're surely going to get far better info than I can give you (old, long retired) but maybe get you at least started.
First suggestion is that it's highly unlikely that the power adapter plug is really "like a stereo headphone jack" (that would necessitate shorting it at least momentarily as it was inserted/removed) So, respectfully, are you sure?
Second would be that you examine the connector with both your reading glasses and the biggest magnifying glass you can get ahold of. Standard practice is to mark polarity on such jacks; but on a wristwatch size device it will be very tiny.
Third, 100 mils at 9 volts seems an awful lot for any battery that could be put into a wristwatch. That spec may be correct, but there must be some current limiting resistor in series.
Just a start.