Why are amines more soluble than ethers in water?
Diethyl Ether can accept hydrogen bonds readily, and you are correct in pointing out it would be better than diethylamine at doing such.  However, diethylamine can also donate a hydrogen bond, giving it two modes of solvation, possibly both simultaneously.  

Also, there is some minor acid-base reactions that may occur forming diethylammonium hydroxide, which being an ion, would certainly be more soluble than diethyl ether.