The What color is the plastic head of your Ignition key? It should be gray in color and not black. If it's not the original key and it's an aftermarket replacement that was cut and programmed for your car, those keys are known to eventually go bad. They'll either stop functioning completely or gradually degrade and quit functioning.
The only way to know what's going on is to get the codes read. A competent locksmith should be able to verify the key and be able to cut/program a new at much less cost than the dealer. Otherwise it's off to the Dealer you go hi-ho and be prepared to pay a princely sum!
Sounds like you might have just a factory alarm and not SKIM then?
Something like the Optional Chrysler EVS II Security System you could buy from the dealer.
When you use your fob to lock the door and arm the alarm do you hear a double chirp or does the regular horn honk?
Also where is where is the red LED located exactly in your car. Is it on top of the dash near the center close to the windshield or somewhere else?