Here Rāma details how she asked for those two boons.

Why did Kaikeyī ask for the kingdom for Bharata if she already had it as a price for her wedding? Because it took place at a very young age and a lot of time had passed in between. So she asked for the two boons which had taken place in between [then and now]. So there was nothing wrong with her asking for it.

Mantharā had asked her, “Kaikeyī, don’t you remember it?” (Rāmāyaṇa 2.9.6) So, since it took place a long time back, it is appropriate that Kaikeyī had forgotten about it.

Why did Daśaratha get ready to give the kingdom to Rāma the kingdom that he had promised to give to Kaikeyī’s son at the time he married her?

Daśaratha considered in his heart the scriptural statement:

udvāha-kāle rati-saṁprayoge
prāṇātyaye sarva-dhanāpahāre
viprasya cārthe ’py anṛtaṁ vadeyuḥ
pañcānṛtāny āhur apātakāni

“One can speak untruth at the time of a wedding, to preserve  one’s conjugal relationship, in a life-threatening situation, when one is about to lose all of his wealth, for sake of a vipra. These five forms of falsity are said to be nondegrading.”

Therefore Daśaratha considered that it was not wrong [to give the kingdom to Rāma]. King Kekaya, being attracted to Rāmacandra’s auspicious qualities, also did not ask the kingdom for his grandson. Rāma had heard about this secret from Sumantra and others. It should be understood that they had known it and [yet] they let things go by [naturally].