What is the angle between two of the nitrogen-hydrogen bonds in the ammonium (nh4) ion?

Answer is: 109,5°.

Nitrogen has eight electrons in the outer level. There are therefore four pairs, all of which are bonding because of the four hydrogens.

The ammonium ion has exactly the same shape as methane, because it has exactly the same electronic arrangement, ammonium ion is tetrahedral (sp3 hybridized.