Ok, I'm sure this is wrong, but so far, this is what I would hypothosise:
2KMnO4 + 14KI + 16NaHSO4 --> 2Mn2+ + 8H2O + 8K2SO4 + 8Na2SO4 + 7 I2
You are right it is wrong - it is not balanced. Not only the mass is conserved (which means atoms must be balanced), also the charge is conserved (so the charge on the left must be identical to the charge on the right). Apparently this is not the case - you have +2 on the right, but 0 on the left. In the net ionic reaction Mn
2+ would be a perfectly valid product, but as you are listing uncharged compounds as everything else, you need to find a counterion to list Mn
2+ as an electrically neutral salt.
Net ionic reaction is the one that doesn't list all spectator species (ones that don't change during reaction). In the net ionic reaction you don't care about exact compounds reacting, but about ions reacting. For example
NaOH + HCl -> NaCl + H
2O
is an overall reaction, while
H
+ + OH
- -> H
2O
is a net ionic reaction. Note that if you mix any strong acid with any strong base counterions (like Na
+ and Cl
- in this case) don't change - before and after reaction they are freely floating in the solution. Net ionic reaction shows what is really happening, ignoring everything else.