The damage control screen in the individual ships window allows the ship to attempt self repairs in the field using onboard maintenance supplies. As your fighter does not store any maintenance supplies, damage control wont work.
To repair the engines (or anything else on the fighter) you can either dock it on a mothership, or send it to a planet with a free shipyard. If you use a mothership, all you need to do is dock the fighter and wait a few hours. If sufficient maintenance supplies are available in the mothership the repairs should be automatically made. If you use a shipyard, make sure the fighter is in orbit over the correct planet/system body and open up the economy window. Under the shipyards tab go as if to build a new ship, but instead select "repair" under the dropdown menu and select your fighter's class and name in the next couple dropdowns.