What I did is the following:
Half Reaction 1 :
KMnO4 + NH3 ----> KNO3 + MnO2
Add H2O on the left side and then 5 H+ on the right, and then 5e on the right :
H2O + KMnO4 + NH3 ----> KNO3 + MnO2 + 5H+ + 5e-
Half Reaction 2 :
KMnO4 ----> MnO2 + KOH
Add H2O on the right side, then 3H+ on the left, and then 3e- on the left :
3e- + 3H+ + KMnO4 -----> MnO2 + KOH + H2O
Half Reaction 1 x 3
Half Reaction 2 x 5
3H2O + 3KMnO4 + 3NH3 ----> 3KNO3 + 3MnO2 + 15H+ + 15e-
15e- + 15H+ + 5KMnO4 -----> 5MnO2 + 5KOH + 5H2O
Then of course, add the equations:
3H2O + 8KMnO4 + 3NH3 + 15e- + 15H+ -----> 3 KNO3 + 8MnO2 + 5KOH + 5H2O + 15e- + 15H+
Net :
8KMnO4 + 3NH3 ----> 3KNO3 + 8MnO2 + 5KOH + 2H2O
I don't even know if you are allowed to do that reaction split up like i did, but if this was the test, that is exactly what I would have done.
..