他启发他的学生 - this one is grammatically correct and can express the idea(but a little bit weird).
他跟他的学生启发 - this one is totally wrong.
---------------------------------------------------------------------------------
>using 启发 as a noun:
我从这本书中得到了启发. - I draw inspiration from this book.
这本书给了我一些启发. - This book gives me some inspiration.
>using 启发 as verb
他的话启发了我. - His words inspired me.
老师要启发学生的思维. - Teachers should inspire the mind of the students
你们要启发学生的创造性 - You should inspire the creativity of the students.
BTW, sentence one in your question is weird , or maybe stiff , coz we do not often use 'sb 启发 sb', we are using 'sb 启发 sb's something' or 'sth 启发了 sb' .