Can I extract water from diethyl ether to get anhydrous diethyl ether?
No it would not work.  Ether boils at a lower temperature than water.  Plus, it forms an azeotrope with water:  a composition of 98.7 % (by weight) ether boils at 34.2 &deg;C and the vapor phase has identical composition as the liquid, meaning that distillation cannot remove the final 1.3% of the water (at atmospheric pressure).

The standard way to remove water is (i) first to use drying agents such as magnesium sulfate to remove most of the water, and then (ii) if needed, eliminating the final traces of water with small amounts of sodium (or potassium) metal.