[noun] + を + 尽くす [noun] + に / のために + 尽くす [verb ます-stem] + 尽くす
尽くす means "to use up completely": to give something everything there is, leaving nothing over. On its own it takes an effort with を – "do everything you can" – or a person or cause with に or のために – "devote yourself to".
The usual objects are fixed: ぜんりょく全力, さいぜん最善, ちから力, て手 ("every means"). ことば言葉につくせない尽くせない, "beyond words", is a set phrase too.
After the ます-stem of another verb, 尽くす means the action has been done to everything there is, down to the last bit or the last corner.
きごう〜きる also means "completely", but it says an action was carried through to the end; 尽くす says nothing is left. 食べきる is finishing your plate; 食べ尽くす is eating everything there was. 尽くす is the more written of the two.尽くす is transitive: someone uses something up. Its intransitive partner is つきる尽きる, "to run out", and the compound form is 〜尽きる.