To balance this chemical equation, follow these steps:

First, let's write down the unbalanced equation:
H2S + KMnO4 = K2SO4 + MnO2 + KOH + H2O

Next, let's balance each element one at a time:

Balance the manganese (Mn) atoms:
There is one Mn atom on the right side, so we place 1 in front of MnO2.

H2S + KMnO4 = K2SO4 + MnO2 + KOH + H2O

Balance the potassium (K) atoms:
There are 2 K atoms on the right side, so we place 2 in front of KOH and K2SO4.

H2S + KMnO4 = 2K2SO4 + MnO2 + 2KOH + H2O

Now balance the hydrogen (H) atoms:
There are 6 H atoms on the right side, so we place 6 in front of H2O.

H2S + KMnO4 = 2K2SO4 + MnO2 + 2KOH + 3H2O

Finally, balance the sulfur (S) atoms:
There is 1 S atom on the left side and 2 S atoms on the right side, so we place 1 in front of K2SO4.

H2S + KMnO4 = K2SO4 + MnO2 + 2KOH + 3H2O

Therefore, the balanced chemical equation for the reaction is:
H2S + KMnO4 = K2SO4 + MnO2 + 2KOH + 3H2O