hello,
here are a few suggestions:
the first and easiest thing to check would be fuses. open up the fuse box and make sure that fuse#5 is ok. if the fuse looks ok , give it a spin in its holder and then try to start the car.
another possibility would be the K38 relay(located behind instrument cluster)
see the following thread,, specifically step 4:
http://www.peachparts.com/shopforum/tech-help/269799-how-disable-alarm-1993-w124.html
if you are positive that the batt. is ok , and you have ruled out the above mentioned, then i believe that your problem may be a bad starter.
good luck