PA-44 student pilot reported returning safely to departure airport after experiencing a bird strike that potentially affected the landing gear and left engine.

Narrative

During the first departure to southwest at 1;000 ft we had a bird strike. We requested the Tower to return back to [departure airport]. On the way back after extending the landing gear; the left main gear position light was not illuminating. We tried to troubleshoot the landing gear following the checklist. We executed an emergency gear extension and requested the Tower to do a low pass to verify the gear position. ATC informed us the left gear appeared to be damaged. For a better visual of the gear; ATC requested us to do a turn back to the opposite direction and do another low approach. On the second low approach; they confirmed the gear appeared to be half down and crooked. So we [requested priority handling] and we turned around again and did another low approach to establish control effectiveness. We requested to leave the airspace and gain some altitude to try to get the gear down by doing maneuvers. We did not have a positive indication of the faulty gear. We decided to secure the left engine because of unknown damage to the left wing due to a bird strike. We landed on runway; the gear was still intact after landing. We vacated on taxiway and shut down the engine following the checklist. We got towed back to the ramp safely. Airplane is grounded for further maintenance inspection.