What atomic orbitals on the nitrogen atom and the hydrogen in ammonia overlap?

The electron configuration for N is 1s²2s²2p³.

The hydrogen atom 1s¹.

So nitrogen requires three electrons to be isoelectronic with neon. Looking at an orbital diagram for nitrogen we note that there are three atomic 2p orbitals that contain unpaired electrons.

So the N-H bonds in ammonia are formed by overlap between a 2p atomic orbital on nitrogen and a 1s atomic orbital on hydrogen.

One very interesing observation must be made. The H-N-H bond angle in water is 107 degrees. But if we predicted the bond angle in ammonia based on the overlap of the atomic orbitals we would say the H-N-H angle is 90 degrees. What's wrong with our prediction? Why is the actual angle so different from the predicted?