first, wait for other users than myself to chime in first, but i'm at least able to get some ideas flowing.
if outputs are blown, you likely have to replace the module. it may be a defect within that unit that caused the outputs to give out. if you repair it, the problem may come up again. also, while i'm on it, a power surge could possible cause similar damage. it should always be used with a surge protector. i will assume going forward that a surge protector was always in use.
if it's still under warranty, i would call Alesis and start the process for replacing the module. then put that thing back together so it doesn't appear to have ever been opened. i have no idea how long they allow people to return defective modules, so even if you think your warranty period has expired, you still may want to contact them to see about getting the defective unit replaced. don't mention that it was opened when you tell them about the popping and the other noises before the audio outputs in question ceased functioning.
i think you'd have a hard time with most of the parts. all depends on where it's fried. you could likely find the 1/4" TRS Female input hardware, depending on which designed is used. you may have to see if another design will both fit and work. if the problem is from the jack to where the board connects, you may be ale to solder it, if it had been soldered in the first place.
as much as i hate to say it, i'm going to bet that replacing the module is the best course of action. sorry and good luck! hopefully one of the other members will deliver better news and info than what i've offered.