"The key provided seems to be invalid. Please check again."

There’s no official app for Wanikani. Judging by the screenshot, is that Jakeipuu?

If so, the official thread for that is here:

For API key issues, check that you don’t have any spaces in the key when you copy paste. The other thing would be to check the thread and see if they’ve added APIv2 support. It looks like the app hasn’t been updated in a while so you may want to use an alternative.

The main recommendations for up-to-date apps are Tsurukame on IOS and Flaming Durtles on Android.