Do you know how to count days of the month?
ついたち (1)
ふつか (2)
みっか (3)
…
とおか (10)
じゅういちにち (11)
じゅうににち (12)
じゅうさんにち (13)
…
Once the day counter becomes regular (11 and on) it ends in にち. Since 何日 is asking what day it is, it uses the same counter you’d use to count the days. Therefore 何日 is なんにち.