All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If you are going to ask the students to leave the academy in order to take private lessons with you and you will tempt them to leave the academy before the contract between them and the academy expires, then this is impermissible. ʻAbdullah ibn ʻUmar narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said, "Let none of you cancel the sale of his brother (by offering a better price/deal)." [Al-Bukhaari and Muslim]
The Ijaarah transaction is the sale of a service. The contract between the students and the academy to take lessons there within a specific period of time in return for fees is a form of Ijaarah. The hadeeth mentioned above applies to Ijaarah just as it applies to sale transactions. Shaykh Al-Islam Ibn Taymiyyah said, "The same ruling applies to trying to cancel the Ijaarah contract of a fellow Muslim with greater reason ... The harms incurred in case of canceling the Ijaarah are often greater than those incurred by canceling the sale. If a man rents a house or works in a place to earn a living and someone else tries to take his place, the consequences would be more unpleasant." [Al-Fataawa Al-Kubra]
However, if the students left the academy and canceled their contract with it or the contract expired, then there is no harm on you in giving them private lessons, especially in light of what you mentioned; that your contract with the academy does not state that you cannot give private lessons to the students.
Allah knows best.