Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas for the expected values of the random polyomino chain to construct the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index. We also carefully examine the highest degree of the expected values for a random polyomino chain through the multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index.