Luis,
I know your jeep is a newer model then mine. Mines a 1995. But, I had some similar problems as you a couple of years ago.

Here is what happened.

It turned out that my constantly draining battery and my dead alternator were symptoms... or, perhaps a better way of saying it.. innocent victims of... a malfunctioning ECM/CPM.

Once I replaced mine. The battery stopped draining and the (replacement) Alternator worked fine.

 

Well the battery is new and terminals in good condition...alternator is new but suspect.After the Pepboys diagnostic report, took the car to a Chrysler dealer for a computer check, They reported That the new alternator was bad and confirmed no repsonse from computer. The PCM was reflashed with no effect. This is not my area of expertise, but in case the reason for getting involved was forgotten, the Chrysler service manager jolted my memory. Alternator about $850.00, Computer about $1200, there could be issues with the wiring after that massive short.....$$$$$$. So far, new battery and alternator $550.00 two sets of diagostics $260.00 towing $200.00 ;;its getting up there.
Called pepboys , spoke to serice manager who said, " I remeber your car, no response from computer, alternator checked out OK." Sooo i decided to run my own tests. Tested alternator by detaching the connector, so alternator was not getting feedback from computer. Connected one output wire to ground and one to positive , theory being, as I understand it, alternator output should be above 13V at the battery terminals. That was not the case. The reading 11.09V which matched the battery voltage, with no change when revs increased, At this point, I removed the alternator and took it back to Pepboys for a bench test, This is where it gets funny....having earlier reported and later confirmed that the alternator was good, they now said that they were unable to check it because that particular alternator was not in their system .........So how did the service department check it? .......They gave me a new alternator. which of course they could not verify was working correctly. The new alternator is now installed, but not plugged in to the system. Before I do connect it to the computer, I should like to know what tests can be run to confirm this alternator is working .any help in this regard would be greatly appreciated. QUESTION: Is there a fusible ling in the charging system?