PA-28 instrument student reported experiencing a rough running engine; which resulted in the Flight Instructor assuming control of the aircraft and diverting to the nearest suitable airfield.

Narrative

While cruising at 6000 ft. ATC instructed me to descend to 5000 ft. I was flying the plane. I am an IFR student and was supposed to take my check ride on Day 0. I throttled back descending at 500 ft./minute. As soon as I leveled off and tried to throttle up to straight and level flight the aircraft began to shake and the RPM's did not respond when throttling up. I asked my instructor what he thought was happening and he immediately put on the carb heat suspecting carb icing. That made the problem worse so he shut off the carb heat. He took out the emergency procedure card and handed it to me telling me he would fly the plane asking me to go through the checklist. I went through the list and nothing seemed to help. We only had about 75% power and immediately informed ATC we had engine trouble and would need to divert to the closest airport. They gave us a choice of airports. My instructor; prior to taking the controls; determined that ZZZ was the closest and we should head there. We were in IMC conditions and could not see the ground. He asked ATC for a vector to ZZZ. ATC gave us a vector and asked us to descend to 2500 ft. from about 4600 ft.. We declined until we got closer to ZZZ. I let him know when we were close and he began to descend while I looked out for the runway. At about 2500 ft. I was able to catch a glimpse of Runway XX at ZZZ. We were instructed to switch to Tower at ZZZ which we did. Tower told us to land on any runway we could. We circled the runway to lose altitude and when able set it down safely on Runway XX.