DA20 Flight Instructor with student reported the aircraft avionics and cooling fan malfunctioned in flight resulting in partial loss of communications. Instructor returned to departure airport with ATC assistance and landed safely.

Narrative

I am a flight instructor and I was flying with a student; practicing maneuvers south of our airport when I experienced some avionics issues that led to a lost comms situation. While we were in flight; an alert popped up on our GPS screen informing us that a cooling fan for the GPS had failed. 30 seconds later; I lost all audio in my headset; including intercom and radio calls. Our GPS unit has the controls for our radios integrated into it. I could not hear anything so I had to remove my headset and yell to my student over the engine noise. She told me that her headset could still receive audio. My student is very new and has less than 10 hours; so I decided I should be the one to talk on the radio. I put on her headset and I was able to hear radio calls from ATC. I decided we should immediately head to our airport due to the risk of further electrical failures or our GPS overheating. However; I made several radio calls to our Tower (a Class Delta airport); but received no response. Eventually; I heard ATC say 'the last aircraft attempting to call is unreadable; I can hear a transmission but cannot make out your voice.' I tried once more and still received no response. At this point I realized that our electrical issue was preventing our radio from transmitting. I still felt it was necessary to get on the ground quickly due to the failed cooling fan; and the potential for any electrical issues to get worse; so I decided to squawk XXXX. As soon as I did that; ATC said 'I see an aircraft squawking XXXX. If you can hear me; press IDENT on your transponder.' I pressed IDENT; and ATC confirmed he saw my IDENT. He gave me instructions to enter the pattern and asked that I confirm that I heard it by pressing IDENT. He gave me a landing clearance and other instructions and each time asked that I confirm it by pressing IDENT - which I did. They told me to expect light gun signals if I did not acknowledge their transmissions with IDENT. ATC was very helpful in instructing us back in to make an uneventful landing.I reported the failure to our maintenance staff at our flight school; and they are investigating what caused the failure.