PA-28 Flight Instructor with student reported engine failure; loss of oil; and smoke in the cockpit during departure climb. Flight Instructor returned to airport and landed safely.

Narrative

Preflight; taxi and engine runup were uneventful and revealed no abnormalities. My student was the PF and performed a soft field take off from runway XX at ZZZ. Roughly 2NM and 800ft AGL; my student said that the engine felt rough; accompanied by a power loss of roughly 200 RPM. I took controls and felt a distinct vibration. I [requested priority assistance] over ZZZ Approach and said that our engine was running rough and that we were turning back to the airport to land. At the time; all other engine indications were normal; and I determined that engine failure was not imminent. Once we completed our westbound turn towards the airport; there was a jolt and engine oil began to spew onto the copilot side windscreen. At the bequest of ZZZ Approach; I [requested priority assistance] and stated my intention to land back at the airport. My student completed an engine roughness and engine failure checklist. I was cleared to land on any surface and continued in what I assume was a glide back towards runway XXL. We were roughly 90 degrees to the runway; so I made a slight turn to line up with taxiway H. Smoke then began to fill the cockpit; so we opened the door and storm window. Due to the oil slick on the windscreen and smoke in the cockpit; I had to lean outside the airplane's open door to see the landing site; which was on a relatively steep incline east and short of runway XXL. We touched down and coasted up the incline to the side of the runway; I announced that we were down and safe to ZZZ Approach and we completed the engine shutdown checklist and egressed the airplane.