The extremely high idle is being caused by the idle air control valve not closing back to the correct position to maintain proper idle. Since the battery was disconnected for a period of time during disassembly/reassembly , the PCM now has to " re-learn" using its own parameters it's programmed with versus actual conditions ( cold temperatures) . You also mentioned the pintle on the IAC valve is " loose " , I'm suspecting damage since it won't reset . The PCM should be calling for it to open for a correct cold engine speed and drop to a normal idle when engine is warm and PCM returns to closed loop . The pintle movement isn't much , no more than 1/8" inch or so . The pintle is too far open and the looseness usually happens from the pintle being moved too far sideways. Repair manuals state a specific installed length and should not be exceeded to avoid damage and adjustment is performed by carefully moving the pintle sideways , back and forth until adjustment is met. Too far and it's damaged. You can try to readjust it to what installed length the manual calls for and take jeep out for a test ride above 30 mph to see if PCM will actually reset it. Be sure that the valve does not open to,far and raise idle too much when jeep is in gear or you know you'll loose control . If a normal idle cannot be maintained , replacement will have to be. Sorry for the bad news. Asuming no trouble codes logged ?
Be sure as well the TPS is adjusted right but that should not cause this problem but will affect trans shifting. Sensors out of range are not helpful to proper operation .
