How can I install a system app that I uninstalled?
Usually, you put the APK back to where you deleted it from. As in your case you didn't check that (but used a backup app to extract it), that "original place" usually is /system/priv-app. Check the structure in that directory (it differs between Android versions), then place your .apk accordingly. After that, don't forget to reboot – as only after that the APK will be loaded.

Note (you know that but others might miss it): in order to write to /system you need to mount it read-write first:

su
mount -o remount,rw /system
# do the copy here
mount -o remount,ro /system


As you have to reboot anyway, you can skip the last step as the boot process then takes care for it.