Kaikeyi opens up her mind 2.11.11-20

Prompted by her partiality for her son and by the joy born of the thought that her husband was entirely under her command, and by that thrice-repeated oath by the king, she fixes him first by asking him to swear by the gods. Doubting that the king may not carry out his promise on some pretext or the other, because it involved the most terrible happening in his life, she binds him firmly to swear by the gods, Sun, Moon, Sky, day and night, earth and heaven etc., She told him: “You have sworn by your dearest son, your virtuous acts and merit. But let the gods hear what you swear”.
Thus preventing that heroic monarch from turning away from his promise, she declared him one who is always truthful.

Kaikeyi specifies the boons 2.11.21-29

The monarch slipped into the snare. Then she spoke, “first, let my son, Bharata be consecrated crown prince with all the preparations now ready for the coronation of Rama. Second: residing in the woods of Dandaka and wearing the bark robes and deer skin, let Rama lead the life of a hermit for fourteen years. I am only asking the boons already granted by you. I am not binding you by any unjust demand. The debtor should redeem himself by paying back the creditor. In that there is no mistake. Let me behold Rama sent to exilement this very day. Supreme as king among kings, be true to your promise and your solar race, to your virtue as well as your high birth.”