You don't need to "implement real controls". Profiler software (like JoyToKey and Antimicro) already does that. The only way to design a game that's "keyboard and mouse only" is to make the game require an entire keyboard, or to require the user to make extremely fast/precise mouse movements.
Your game does not require those. It uses a small number of controls. Therefore, it is already controller-compatible. The only problem is that the current version *forces* its own controls, which requires a trial-and-error struggle process to decipher. If the game didn't impose anything, it would be immediately friendly with JoyToKey.