My first guess is the IAC is not regulating the idle air flow or there is a vacuum leak in the intake tube between the MAF and TB. This could also be a weak cylinder(s) not pulling their weight. How does the car run otherwise?
How long has this problem been going on? I ask because after a battery disconnect, the PCM needs time to re-learn new idle trim values. Has the PCM been given enough time to learn?
Confirm +12 volts to the IAC red power wire key on. If not, there is no power to the IAC.
Do you have access to an ODB2 scanner that can monitor operational data? Would help to know what the IAC duty cycle (percent) is.
For more information on how the troubleshoot IAC problems, check out post #3