I've repaired quite a few of these things and every repair tends to be a bit different than the one before but I'll give you general ideas. You will need to have the air scoop off and all the parts below it to the point that you can see the heater flaps and heater core.
Once this valve is really stuck it can be
very difficult to remove. Sometimes I try to tap it down slightly with a punch so that it breaks the bond between the valve and the outer casing. Once I move it down, I sometimes use a long screw and thread it through a number of small washers placed on top of the casing so that the screw can thread into the hole in the top of the valve. By turning the screw and changing the number of washers used I can sometimes get it to work like a gear puller. Regardless of method, limiting the amount of force is critical. As you have already found out, the screw can break off and then you're ( pardon the pun ) screwed.
Once you are able to get the valve to move even a small amount it will usually come out. Polish the valve so that it will fit into the heater core without binding. I use liquid silicone to lube the valve which also makes the O-ring on the valve work much better.
Mark the angles ( fully open and closed based on where the hole is pointing ) on the top of the valve as it opens and closes ( this is important ) so that you can determine where it's pointing to as you try to index the valve. I concentrate on making sure it's fully closed when the heater lever is off so that there's not any heat for summer use. If it's set up right, you should be able to get very warm heat when it's fully on. You may have to take the screw back out and move the valve one position on the square of the valve ( left or right ) so that all of the linkage works properly when open or closed.
It's not the job from hell but it's pretty close sometimes.
Oh, and don't be surprised if you drop that little piece that slots into the casing that indexes the valve. I have many times and you won't find it either. Best to place some rags or something like that around the core to prevent that from happening.