Translation:Двадцять один день
In Ukrainian as spoken in Ukraine, if the numeral consists of several words, the last word of the numeral determines the form of the following noun. So, in «два́дцять оди́н», you look at the last word: «оди́н», and choose the noun form according to the last word — «два́дцять оди́н де́нь» (see discussion about «Два́дцять оди́н студе́нт» for a more complete explanation, or just google «іменники з числівниками»).
"Двайцять один день." is not correct.
It's not correct in your dialect, but it's correct in literary Ukrainian. Please keep in mind that your Ukrainian is not the only variety of Ukrainian, and it's not the variety taught here. This course teaches literary Ukrainian, the standard language as used in Ukraine in books, mass media, and as taught in schools. In literary Ukrainian, «два́дцять оди́н де́нь» is correct, and «два́дцять один днів» and «два́йсять оди́н днів» are incorrect.