Equation 1.6.25 was derived by considering the effect of the commutator on an arbitrary position eigenket, 1.6.26 is derived from this result by using the definition of the translation operator eq. 1.6.20.

I plugged that into (1.6.25) to work out the commutator. Sakurai claims it is [tex]d\textbf{x}'[/tex], but as you can see, when I work it out, I do not understand how that claim is true.

It's true because of eqs. 1.6.23 and 1.6.24...I don't understand the source of your confusion here. If you agree that eq. 1.6.25 is true, and also that 1.6.20 is true, then 1.6.26 must also be true...it is basic logic.