Flight Instructor reported taking evasive action to avoid a collision with twin engine aircraft that was departing the pattern. Flight Instructor landed safely at the airport.

Narrative

My instrument student and I were on an IFR training flight; nearing the end of our lesson and making our descent and approach back into our home airport. We were on with approach control; who was descending us while providing vectors. After advising the controller that the airport was insight; we received a final vector. Not too long after; we were given our clearance for the visual at 4500 feet and told to contact tower. This set us us nicely to make a left base turn for the runway. We switched frequencies and checked; initially with no response from tower. The tower controller was extremely busy and handling priority calls that were constant over the frequency. Moments later I noticed an aircraft on our ADS-B display that originally was much lower and within the lateral boundaries of the airspace was showing a flight path eventually to be conflict with ours; and the vertical separation between us was shrinking fast according to our traffic display. I immediately took controls from my student to level off and cease our descent while visually acquiring the other aircraft. The aircraft departing the area was a high performance twin engine aircraft that I am unsure of the type; but am certain had significantly greater speed and climb performance than us. I initiated a full power climb while I had a visual on them off our left side. They continued to climb up towards us while we climbed as rapidly as we could away from them. It seemed to me that the other plane was just climbing and climbing closer to us. Ultimately they passed underneath us; at my estimate with vertical separation of 300 feet. Just prior to the aircraft passing underneath us; my student and I got a response from the tower controller instructing us to continue for the runway and gave us the traffic advisory. We reported them in sight; and after passage; proceeded as instructed and started to descend again. I attribute this event to a few unfortunate and untimely circumstances. Both us and the other aircraft and are responsible to see and avoid; and I don't doubt that we each did our best to fulfil this responsibility. We were cleared for operations that put us at conflict in congested airspace used for both arriving and departing traffic. The closure rate of the other aircraft limited the time I had to visually acquire them and make the appropriate evasive action. I don't know if the other aircraft had us in sight; but it did not appear that they ceased their climb while our flight paths grew closer. Our handoff from approach to tower was at a time where this traffic advisory would have been helpful; and the congestion on the tower frequency made the handoff and response delayed. I am thankful for the ADS-B which helped me recognize the potential for the developing hazard early; and that we were able to visually acquire the traffic soon enough to maneuver in a way to maintain separation; albeit much closer than I am comfortable with. I think the difference in aircraft performance played a role as the traffic was on a rapid and fast climbout; developing into a collision threat very quickly. Our only option to climb away from the conflict was much slower than the climb rate of the other plane. My biggest takeaways from this event and what my student and I de-briefed on consisted of the importance to always see and avoid; to be weary of congested airspace where aircraft are departing and arriving; be cautious not to develop a reliance on ATC providing separation even when in radar contact. At the end of the day; we are responsible to see and avoid; and did our best to do so.