For 静か and 暖かい, the か is a fossilised grammatical element (cf. 静まる and 暖まる, which don't have it).
As for 大きい, it's written with き to differentiate it from 大い, which is a 形容動詞 (albeit with a similar meaning). 小さい's case is a little less clear, since while there is a word that's written 小い, it's a very informal word (ちっこい) typically written with kana. It may be out of a desire to match up with 大きい.
There's a few other adjectives that have extra okurigana for differentiation purposes as well (eg. 危ない and 危うい from each other, or 少ない's archaic 終止形 少なし from 少し).
The fact that both of these adjectives (大きい and 小さい) have irregular 連体形s (大きな, 小さな) may also contribute to their unusual spelling. Certainly 大きい has an odd history - it started off as *əpəsi, 連体形 *əpəki, just like any other adjective; but in the Heian-jidai it started being used as a 形容動詞 as well (of the form *owoki (nari)). This in turn led to its root being reinterpreted from owo- to owoki- in the Muromachi-jidai, and the -(k/ɕ)i just ɡot stuck back on the end to form owokiɕi (> modern 大きい, via the 連体形>終止形 shift).