PA-28 instructor pilot reported returning to the departure airport after experiencing smoke from overheating electrical wiring.

Narrative

Upon receiving an IFR clearance on the ground I took off with a student to do local instrument approaches as part of a training event for my student's instrument rating. At 1700 ft. msl or between 900 and 1000 ft. AGL the student called out smoke in the cockpit; immediately after I could smell it and see it in the cockpit. It was light gray in color and smelled of burning insulation/wires. We ran part of the procedure listed in the POH for the emergency venting the cockpit of smoke by opening all the air vents however we were unable to shut off all of the electronics per the procedure in the airplane as we were now in IMC and needed an instrument approach to get to the ground and were flying a PA-28-161 equipped with an Avidyne Integra electronic flight instrument system. We informed Approach Control of the issue and stated we would need to return to the airport and were therefore continuing to head to the initial approach fix for the ILS approach. [ATC] asked us if we [required assistance] and I said yes. They gave us a prompt approach clearance in to the airport and got all the standard information such as fuel on board; number of souls on board; and color of aircraft. We successfully completed the ILS and once on the ground cancelled our IFR clearance and reported what time we reached the ground to [ATC].In the end I've identified that the major risk in the situation as far as human performance goes is the high number of tasks that's necessary to complete when going immediately from the takeoff and climb phase of a flight to the approach phase of a flight and this task level is ripe for human error to be present. One mistake I made on my part was not [requesting assistance] in the initial notification of smoke in the cockpit to [ATC] and they had to ask me if I [required assistance].