Well, it's your Heater Core (or heat exchange).

I just replaced mine, and while not being extremely difficult, you have to take apart a lot of stuff to get to it. (thank God the access point is at the front of your car and not under your dash!)
You have to remove:
battery plastic covers
wiper's arms
plastic cowl
battery
wiper mechanism
stress arms/bars
center bracket

then pinch the two coolant hoses with the correct tool
remove the hoses
remove the foamy top cover
remove old heater core from housing
clean all the mess (do you know where all those pieces of foam are coming from your vents??? yes, from that flap!)

Re-instal in the correct order.

To minimize bubbles on your coolant system, pre-fill your core with fresh coolant.

After installation, release the purge valve on the coolant tank and turn on the car. check for leaks and keep an eye on the level.

No more coolant/musky smell inside the cabin, no more dripping on top of my shoes and the heat now works.

 

Good write up, any pictures?

 

Didn't have any luck with the camera. Had a little issue with one of the clamp pinchers and had to run with the car semi-disassembled to buy another one... not bad but very laborious...