I completed the mission all three times, but it is still in my mission list and drones are still showing up on the battle tab.

Did you just get three drones and click three times? You need to find a total of 90 drones and click the mission button 90 times. It costs 9000 energy, I believe. How many T-Plasma Gemini Cannons do you have? You should have three. Check your cargo and your ship. Lastly, if you're still 100% sure that it should be gone by now, send an in-game report to Dan.