My wife dropped an iPhone 4s which I opened. I found a small piece hanging lose, near the top of the battery. Front side...

If it has been a few weeks, it is unlikely an effect of the juice spill. Juice will dry, but not at such slow rate,so any effect would be immediate or latest after a day. Your problem sounds more like a damage, such as an intermittent connection or solder point. Try to just push and pull the plug, without pulling out the plug, to see whether my theory is right.

Microphones convert mechanical movement (membrane) into a small voltage (inverse function of a speaker), so water likely went inside, and now the membrane does not vibrate any more and hence the microphone does not work. If it was salt water, it is likely corroded and will be damaged mechanically. If it was sweet water, you may be lucky and after it dries it may work again. You could try using a hair blower to dry it. If that does not help, there may be some small particles inside preventing the membrane from vibrating. Probably nothing much you can do about that.