Can I say that Language Identification is part of Text Mining? If yes, what is exact relationship between these two terms?
Yes you can! Language identification (LI) is a pure multi-class classification problem, while text mining (TM) can be understood as a family of concepts (including text classification, clustering and other machine learning but also natural language processing concepts).
BTW: LI can be seen as a (near)solved task, as existing approches achieve very high accuracies (to my best knowledge around 95-100% even for exotic languages). However, this does not hold in general for TM, as it highly depends on the involved tasks.