Approach Controller reported an aircraft did not fly the published missed approach and the pilot reported encountering severe icing conditions and several equipment malfunctions/failures while in IMC. Controller vectored pilot to another airport and they landed without further incident.

Narrative

At XA:XX Aircraft X requested and was cleared for the GPS RWY 14 Approach into 3D2. I (the approach controller) obtained the weather at 3D2 and advised the pilot that there was 1/4 mile visibility and 002 ceiling with fog. They told me thanks for the update and continued with their approach. Just inside the IAF they were switched to advisory frequency and told to report IFR cancellation when able. Upon their missed approach; the aircraft did not fly the missed approach procedure; but instead went under their own navigation and attempted to land at 3D2 for the next several minutes. Approach control attempted several times on frequency and guard to contact the aircraft and obtain their intentions. Finally; when they were unable to land; the pilot climbed to 030 and contacted approach; requesting a GPS approach into ZZZ. Approach informed the pilot that ZZZ also had ceilings reported below GPS minimums and requested intentions. The pilot chose to divert to ZZZ1. The Approach controller issued a vector for the GPSXX into ZZZ1; and shortly thereafter noticed that the pilot seemed to be unable to maintain a vector. The Controller inquired into the condition of the pilot and the aircraft; to which the pilot then reported that at 3D2 they had picked up severe icing and had several equipment malfunctions/failures. The pilot went on to say that the equipment had all come back and they had come out of the icing; but they still had a 'frozen manifold pressure gauge'. The controller asked if they were in an emergency situation and required assistance; to which the pilot declined and said they were able to continue without assistance. The pilot was then told to call GRB TRACON when on deck; to talk more about the situation. The pilot landed at ZZZ1 at XB:XX without further incident; and called at XB:XY. After asking about what happened; the pilot told me that from the point where they began their approach into 3D2; they had begun to pick up ice; and by the time they were overflying the field; they had 'lost everything'. They attempted several times to land; but could not get the field in sight; and eventually climbed and came back to me. Suggestion: I would like pilots to report any unusual or unsafe condition as soon as possible (i.e. icing as soon as they began their approach; major equipment failure) and also to monitor guard if able. The weather reported at the field was well below minimums for the type of approach attempted. I should have issued more a warning against attempting the requested approach; instead of just issuing the reported weather again.