Looking at your photo, I would grind the rust off of the mating surfaces, reposition the hinge tube with the door in place, clamp the hinge tube tight to the stove and stick or mig weld the hinge back in place, then open the door and weld the back side of the hinge. It would seem an easy fix.
Arc Tourist - I have cleaned it up in the past, that is just very light surface rust and will be taken off before welding.
I meant, what type of rod or wire....will it need to be a suitable cast welding rod or will an ordinary mild steel rod or wire do the trick? The top of it is welded, so using other information i've read on here, it will probably be made of cast steel?
It will, im sure, just be mild steel and then the hinges will be tube that the manufacturer has cut into sections and welded on, badly, if you've managed to remove one.
I'd suggest that it's just a standard welding job to sort it out.
I'm a novice welder though, so take anything I say as nonsense... : )
Here is an alternative solution to consider. Get some mild steel tubing and angle iron which you can expect to weld successfully, and make up a new hinge attachment which can be bolted or screwed to the log burner. Here's a rough sketch.
the log stove looks to be steel made cant really tell with the poor quality pics
the bar on it looks so out of place it dosnt look like it belongs and i cant decide if its rust on the bar or a copper insert on the hinge which would suggest more bodging
sorry to be blunt here dosnt appear to be a model im aware of however i cant seem to throw the ref that its something sold on ebay
as goes for welding it would be an arc welder for steel cast iron id be looking for another way
cast stoves normally go on top plate by over heating them or on there grills
Although the pic's slightly out of focus, the top hinge knuckle certainly seems to have been welded to the door frame.
A quality cast iron stove would almost certainly have had the knuckle cast in to the door frame, a cheaper (Chinese one?) would probably have the hinge bolted/screwed to the casing - (as it is with our own Olymberyl model). It functions well enough but could hardly be described as either aesthetically pleasing or sound design.
Getting adequate penetration when arc welding a cylinder to a flat surface (without slag inclusion) is often a challenge, (a 'D' section knuckle as opposed to a cylindrical section is much easier) but mig should be fine.
Clean the area and the knuckle of rust, clamp it in place making sure the hinge pin will pass through it and the door closes correctly, weld away.