I've seen similar situations (although not with 4cm of icing!) in situations where there doesn't seem to be any leaking of exterior air, but the interior humidity was quite high in a localized area of freezing. Simple answer: consider dehumidifying, or venting to interior air, the affected area. [Since there is no further work to be done with the door, according to your post.] Heating the affected area may also work, but I'll explain why I consider this a last resort.
It sounds as though you have an entryway (or foyer) enclosing this entry door, for practical purposes of airflow (or limiting air flow). If you are using this area as a "mudroom" or area where wet/cold/snowy coats/boots/outerwear can be left or changed from, then the moisture brought in by that wet material could be evaporating, then condensing on the sub-freezing surface of the door, then of the ice as it builds up.
Removing this moisture from the air will assist in limiting the amount of icing occuring. This could be done by any combination of the following:
- Remove clothing/outerwear/footwear that has moisture from the affected area after initial entry. Moving that snow-covered coat in from the foyer will mean that it dries inside the main dwelling, instead of in the foyer. This could be drips/puddles in the house, but this evaporating water will also humidify the main living space, and not condense/freeze on the entry door. A rubber mat/tray can also help with this. Footwear can be removed to bang off loose moisture on the exterior of the dwelling, then set to dry in the living space.
- Dehumidify the entry. A simple dehumidifier can do this, or simply open the door to the outside/inside for a few minutes each day. The air exchange (although wasteful of conditioned air) will also exchange the moisture in the air. Depending on the amount of moisture, however, the air exchange may need to be done multiple times daily. On the up-side, this air exchange, if done to the interior, may warm the space enough to thaw the ice off of the door itself, giving a "reset" of the situation.
- Heat the entry. I'd consider this a last resort, because heating will not solve the problem. The problem is a sub-freezing surface, which you will not be heating - the air surrounding the sub-freezing surface will be heated, and the heat will be quickly conducted through the thermally-efficient conductor of the door surface to the exterior of the dwelling, once again returning to the state of maintaining a sub-freezing surface on the door.
Provided there is truly no leakage of air, this is actually a humidity issue. It may be worth it to buy an inexpensive (but accurate) humidistat to check the humidity levels. The tool would have future uses, so it's not a one-time purchase without future utility. Lowering the humidity level in that space is the best way to solve the problem.
For the future
In the future, (or perhaps this can be done in the short-term as well,) consider treating the interior surface of the door. A non-stick spray (silicone lubricant) or wax (for skis, cars, even just regular candles) would help the moisture (and ice) be more easily removed from the surface. If the condesation cannot be stopped, it could at least be mitigated by quick removal when someone passes through the doorway. 4cm of ice isn't good, but 1cm of ice that falls off everytime someone slams the door closed coming or going may limit the thermal compromise of that room, and ease in operating the door. There's also the added bonus that ice is solid moisture - so dehumidifying the room of that chunk is a simple as throwing or sweeping it out the door.
A long shot
There is one other option, but it's not necessarily adviseable. Venting the room completely to the outside or inside of the dwelling will provide constant air circulation, especially if induced via convection. This would circulate the higher-humidity air with the larger vented-to area. However, this would provide a heat-sink if vented to the inside, and/or a very cold room if vented to the outside. (You may also transfer your freezing problem to an interior door if ventin gthe room to the outside.)