My first reply was based on the title of this topic. Your last suggestion seems to say you had it working. So you did have a porperly installed Android SDK.
Another thing, after reading your message: How do you emulate an Ionic app in Android Studio.
IMHO you cannot run an Ionic app in Android Studio.
As far as I know to emulate an Ionic App your emulate or run it from the terminal with:
sudo ionic emulate/run android
For emulate you need an Android simulator.
For run you need a real device connected to your PC.