The dragon is less helpful than you think
the only better materials for making swords than medieval steel is modern precision steel alloys. The main problem medieval smiths had was they could not reliably make uniform steel, one end of a billet might be pig iron while the other was nearly pure iron with dollops of slag throughout. If the dragon can precisely control its heat that is help, but if he does not have good steel to work with, that is uniform steel alloys, he will just be rolling the same dice medieval smiths had to put up with. To make high quality steel the dragon needs chemistry and precision control of temprature. First he needs good steel which is down to chemistry, that is the biggest advantage your dragon could offer but nothing about being a dragon will help here. Of course you can handwave the dragons knowledge and get whatever steel you want, but I am working under the assumption the dragon has top of the line medieval knowledge and nothing more.
Properly hardening blade steel is about hitting a sweets spot of temprature and cooling. See my answer about dragon blood quenching if you want an idea of how to get a real advantage. The main flaw medieval swords had was the inconsistency of the metal. They tending to have the quenching down fairly well. Again precision control of temprature will help here but only if they have good steel to start with. Medieval smiths were not limited by heat but by their knowledge of chemistry.
Use a normal sized sword, make it stronger instead. Once the blade of sword gets longer than about a meter you start seriously reducing your range of motion with the sword, real swords did get past this but many of those were single purpose swords, often anti-Calvary or functioned more like spears than traditional swords. You can make the sword a little heavier to make it a bit stronger (3 kg is a crazy heavy longsword but should be usable by your hero) but you are limited by three factors. First no matter how strong your hero he is still working against physics, bigger sword equals slower sword for the same strength. It also quickly means less controllable sword becasue the sword weights more compared to the hero's mass. When you swing a sword the sword also swings you, the huge difference in mass is the only reason it is hard to notice. No matter how strong you are the friction holding your feet in place is still the same. Second you don't want to make the sword too thick as that reduces its ability to cut.
Third Making the sword bigger also does not keep it from breaking, because the tang is still the same size, even with the best case scenario "the entire handle is solid steel", you can't actually make the sword that much bigger before the handle becomes the weakest point on the weapon. Using a solid metal handle will also suck for shock transmission. Giant sword with human sized handle ends up being weaker than normal sword with human sized handle. Picture trying to swing a cartoonish cardboard sword, the handle gets floppy first because it has the smallest cross section of material.
More importantly you are never swinging a sword as hard as you can. If you want to hit a static target as hard as you can use an axe, that is not what a sword is for. Sword fights are won or lost via speed, strength is rarely an issue, when it is the strength of the sword is largely irrelevant, it does not take much strength to kill someone with a sword. Super strength does have a benefit however, it means he can swing a sword much faster than his opponent and even strike from unconventional positions, which is a much bigger advantage than swinging a heavier sword at a normal speed. Also if your hero is routinely hitting steel with the blade of his sword he is using his sword wrong, of course that was practically the only the advantage of a double edged sword you have a spare edge once you have dulled one.
Using the sword.
Your hero is levering his strength into speed, because of his strength he is capable of moving himself (although he may have to invest in some cleats to move himself much faster however) and his sword faster than his opponent and is capable of striking from unusual positions. You almost never swing a sword with your full strength, truthfully your grip on the sword is were a person hits the limit of their strength first. Both speed and unusual strikes will give him a distinct advantage over other swordsmen and speed will give him a big advantage over anything else. Again you don't actually have to swing a sword that hard to kill someone, the hard part is getting it into the right position.