A bit of history of when and how this first happened might help, it may just be that the main fuse for the alarm unit has blown.
Do you know where the alarm control unit is fitted ,if it's a Cobra it's not an original fitment and could be anywhere, they are fitted in a location where tools are required to gain access for security reasons.
You are right about the key in the siren ,this only de-activates the siren.
Really your best bet is to get a car alarm fitter to have a look as they know the usual locations on all cars, or your local car thief .
Geo.