C172 Flight Instructor reported executing a go around to avoid a potential collision with a King Air landing on an intersecting runway.

Narrative

I was training a pre-solo PPL student in the pattern at ZZZ about 20ish minutes before sunset. There was constant light turbulence; occasional moderate turbulence; and a slight left crosswind which made for good light crosswind/turbulence training. A king air reported that they were inbound on the RNAV A; and eventually made it clear that they would circle for runway XX. We were using runway XY since it was the favorable runway and also since there were other aircraft in the pattern that were using XY. My student and I were either in the downwind or on base for XY when we saw the king air breaking off of the RNAV A and entering a sort of modified crosswind for XX. Thinking that we had ample time we continued to fly our pattern and call each leg. We turned base to final and I saw the king air in the downwind seemingly just past the numbers. Still thinking we had ample time; we continued our approach. They turned their base seemingly very early and seemed to continuously turn into final without leveling off on the base leg. When we were on short final we made a short final call to let them know where we were; and then they called short final. They were still much higher than us so we continued with the landing. We dipped below the hangers and I lost sight of the king air and I focused on the students landing. As we were getting close to the intersection and passed the view-blocking hangers I looked to my left and saw that the king air was either about to flare or in the flare. At that point we were just about to pass through the intersection and I pulled up on the yoke to gain altitude as quickly as possible. We proceeded in the upwind and the king air rolled by behind us. When I first saw the king air in their crosswind; I thought that the timing would work out perfectly; that we would enter our upwind as they were towards the end of their downwind. Even when I saw them when we turned base to final; I thought we had more than enough time for safe separation since they didn't seem that far along in their downwind and since large multi engine aircraft that operate in ZZZ usually fly a long and wide pattern. However they turned base fairly early and didn't seem to come out of the turn until they were on final so they were much quicker than I anticipated. We also had a large headwind on final so that likely made us slower than I anticipated. All of these factors made the ample separation that I though we had get smaller and smaller.I had thought about going around; but thought that they were more likely to go around since we were lower and if we both went around then we would still be on the same course. After we had touched down; I thought about making it a full stop and trying to come to a stop before reaching runway XZ/XX; but I decided to add power and go since I wasn't sure we would be able to stop before the intersection and if we slowed down and still passed into the intersection then we would have lost time and that would further increase the risk of a collision. I should have circled when we were on final; added power and broke away from the direction of runway XZ/XX.