Google stopped updating the core framework for Android 4.4.2 several years ago. The "latest" version of Play Services available today requires Android 5.0 (Lollipop) or higher. If you attempt to install a modern APK on a KitKat device, you will likely face a "Parse Error" or the app will crash immediately upon launch.
Do you need help (like ARM or x86) for your device to pick the right file? Google Play services (Android 4.4+) APKs - APKMirror Google Play Services Apk Download For Android 4.4 2
Since you cannot install Google Play Services via the Play Store (because the Store likely isn't working), you must enable . Google stopped updating the core framework for Android 4