GA Flight Instructor reported a near miss with another landing aircraft while landing on an intersecting runway.

Narrative

I was conducting an aircraft type checkout flight with my friend who was occupying the front (primary flying) seat as we entered the pattern and landed and Coeur d'Alene airport (COE). We had determined there were several airplanes operating in the left traffic landing pattern for runway 20. We announced our intentions on CTAF and entered a left downwind via a 45-degree entry. We arrived on the downwind with 2 aircraft in front of us; one on base and one on final for runway 20. In an instructing role; I was dividing my attention between monitoring the traffic and coaching the student on landing procedures for this aircraft. Another factor is that the wind was reported at 140 at 9 which is a left crosswind for runway 20; and as this is a a type of aircraft; that creates a situation that requires additional attention and focus. At some point on the downwind; we heard an airplane report on an instrument approach for runway 06. I was focused on making spacing with the 2 aircraft in front of us for runway 20; and on the landing procedure; and at the time I judged the instrument traffic for runway 6 was a long way out. As we aligned for final for runway 20; I lost situation awareness of where the runway 6 aircraft was. We had been announcing our position and at one point I vaguely recall an aircraft transmitting something about rolling to the end of runway 6. For some reason this did not register in my mind as a potential conflict; thinking; perhaps he was still on several mile final as we approached short final. As I was focusing on how the student was getting ready to flare for landing and align for the crosswind; I heard another statement about someone turning off the runway at the end. Another airplane on frequency made a transmission with something of the effect of stating to use caution as there was an airplane (us) on short final for 20. As we passed the intersecting point of the runways; still 100 feet or so in the air; I looked over and saw Aircraft Y; at taxi speed; 500-1000 feet back from the intersection on runway 6.I was frustrated with myself for loosing situation awareness of the traffic situation at the non-towered airport with intersecting runways. The factors mentioned above contributed to a task-saturation situation that I should have managed more effectively. This was a personal reminder to make effort to better manage my attention while instructing in a busy traffic pattern. The issue of traffic separation challenges at this particular airport (COE) is already being addressed with a temporary control tower that is set up and operational during the summer months. Many pilots have suggested the traffic volume; intensive flight training operations; and intersecting runways at this airport contribute to justify consideration of a full-time tower.