尽きる is "to be exhausted." 尽くす is when you're doing the exhausting of something, which makes it to exhaust something. From this, it can also mean to devote oneself, since you're probably using up every last bit of effort you've got when you're devoted to something.

You'll also see 尽くす attached to other verbs to mean to do fully, like 使い尽くす (to use up completely) or 焼き尽くす (to burn to ashes).

This uses the same reading as 尽きる, so you should be fine here.