C208 flight crew reported encountering severe icing conditions that had overwhelmed the anti-icing system on the aircraft. After landing; significant ice was discovered on the airframe along with one inch of mixed ice on the leading edges of the wings.

Narrative

Prior to leaving ZZZ; First Officer (FO) and I checked current weather observations and forecasts en route and for ZZZ1. There was no icing PIREPs and nothing was forecast near the airport. There was light icing depicted on the current chart on the weather website near the ZZZ airport at 3000 - 5000 ft. Based on PIREPs; cloud cover on the iPad; and our inbound flight; we determined that the cloud deck was only a thousand or so feet thick and not a concern for this flight. FO and I verified our TKS fluid was still full since we had just added fluid in ZZZ1; and briefed icing procedures and the flight profile. As we began descending for the RNAV XX approach; I charged the TKS system prior to entering IMC; and it coated the wing with fluid. We entered the clouds at about 4800 ft. I began noticing some trace ice and I activated the TKS system again to 'Norm'. Since I was pilot flying; FO was watching the wings and air frame; and called out light icing. By the time I was able to look out my window; the left wing leading edge was covered with about a half-inch thick of rime ice. This ice was slowly shedding on the outboard part of the wing; but the inboard icing was still built up. I asked FO to look at the tire; the wing; and tie down hook and tell me what they were getting and they advised me of an icing buildup everywhere. At this point as I was preparing to level from the descent; we hit SLDs (super cooled large droplets) which looked like a water balloon hit the windscreen. The water spread across the windshield and froze; obscuring my view outside. I activated TKS max flow air frame and windshield TKS fluid. Even on high; ice kept accumulating. I told FO to let ATC know we were in severe icing; which they did. ATC asked if we wanted to descend below the approach altitude and I said no because we were still IMC; and [requested priority handling] direct vectors to the final approach fix. ATC gave us a heading and I proceeded to turn. FO ran the remaining checklists while I verified the aircraft configuration; I disengaged the autopilot and began flying the course intercept; and pointed out our airspeed. We had been cruising at 147 kts. and were now at 143 kts. As we began descending on the glide path; our flaps-up airspeed further decayed to 130 kts. as the fastest we could get. I introduced flaps 10 degrees and they were introduced with no noticeable change in flight characteristics. There was no wind or turbulence of note. Inbound from the final approach fix; the aircraft rolled un-commanded to the right; left; and began shaking. I cycled the prop quickly to shed ice to 1600 RPM and back full. I crossed the threshold at 119 kts. I was able to land firmly on the 1000 ft. markers and only after landing turned off the TKS fluid. As we pulled into the gate there was still significant airframe icing and about 1 inch of mixed ice on the leading edge of the wings.