Center Controller reported an aircraft had a heater problem; aircraft was icing up; and could not hold altitude.

Narrative

[I was] training on Sector X when the Controller-In-Charge (CIC) told me there was an emergency at [Sector Y] and my pilot skills might be needed. I immediately terminated training and proceeded to Sector Y; where I plugged in with the Radar Controller and offered to assist. The previous sector had coordinated a malfunctioning heater. However; it was quickly apparent that we had a small aircraft that was icing up; at Minimum IFR Altitude (MIA); and could not hold altitude. Over the next 45 minutes; the Radar Controller; CIC; Operations Manager (OM); Radar Associate; Center Weather Service Unit (CWSU) meteorologist; and myself worked the aircraft north and west toward lower MIAs and better weather. On several occasions the aircraft entered turns and descents without instruction to do so; and each time the Radar Controller provided simple; pertinent instructions that helped the pilot regain/maintain control of the aircraft. Another Controller team came back and took the airspace and other frequencies; so our team could concentrate solely on assisting the emergency aircraft. The aircraft was below MIA the entire time; and with no obstructions depicted; I obtained a sectional chart and spent most of the emergency tracking the aircraft on the chart and calling out obstructions to the Radar Controller; who relayed that information to the pilot. Eventually the aircraft dropped out of communication and radar coverage; so we placed another aircraft on the frequency to act as a relay. Wheeler-Sack Airbase also called several times with position updates; as their radar could see him while ours could not. Eventually we were able to get confirmation that the aircraft had safely landed. We only then discovered that the aircraft was a small aircraft [type X]; not a small aircraft [type Y] as the flight plan indicated.This incident happened because the aircraft flew into icing conditions. Our job was made significantly more difficult by the lack of depicted obstructions on our video maps.An Emergency Obstruction Video Map (EOVM) needs to be created for the facility.