It will be easier to change the headphone jack for a new one, if you know someone who could do that.
You ask : "Is there a way to have the sound going into the microphone jack be routed through the amplifier?"
The sound is coming from the amplifier, but for the microphone you have a capacitor and a resistor who is higher than the one for the headphones.
This is the reason the sound is weak, until you change the resistors, you can't change the strength of the signal from the audio chip.
Thanks, looks like I'll either have to risk it and do it myself or just deal with it.