I have developed an Android app with Kivy, which I'd like to publish in Google Play. Now, I'd like to use certain MIT-licensed data, which will ship as part of the app. Is this legal? As far as I understood, I need to include the copyright notice, but I do not understand how to do it. Is it enough to just put the MIT license text inside the 'about' screen of the app?
MIT-licensed data is a bit of a weird case, but yes, showing the MIT license text would be all you have to do for compliance with the license.
The MIT license is very much intended for software, not for other copyrightable works. Data by itself is not copyrightable, but the data might represent copyrightable creative works, or the data may be a database and your jurisdiction recognizes database rights (e.g. the EU but not the US). But lets ignore copyrightability of the data and assume that the MIT license is meaningful and valid.
All the MIT license requires from you is that “The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.” For mobile apps, including these notices in your app's about-screen is indeed the common way to do that. For example, you might have text as follows:
ExampleApp is Copyright 2021 Example Author, all rights reserved.
ExampleApp includes various third party material:
Some Data Set
Copyright 2018 Database Author
Permission is hereby granted [… MIT license text]
If in doubt, a good way to do it can be to emulate other apps that are probably compliant. In my experience, web browsers such as Chrome or Firefox do a pretty good job of displaying open source licenses properly.