1. **State the problem:** We need to find the measure of angle $\angle GKL$ given the expressions for angles at points $K$ and $L$ as $(8x + 11)^\circ$ and $(7x - 41)^\circ$.\n\n2. **Analyze the given angles:** Since the problem involves lines and angles intersecting at points $K$ and $L$, and the question asks for $\angle GKL$, we assume these angles are related.\n\n3. **Use the fact that angles on a straight line sum to $180^\circ$:** If the angles $(8x + 11)^\circ$ and $(7x - 41)^\circ$ are adjacent angles on a straight line (as implied by the diagram description), then:\n$$ (8x + 11) + (7x - 41) = 180 $$\n\n4. **Combine like terms:**\n$$ 8x + 7x + 11 - 41 = 180 $$\n$$ 15x - 30 = 180 $$\n\n5. **Solve for $x$:**\n$$ 15x = 180 + 30 $$\n$$ 15x = 210 $$\n$$ x = \frac{210}{15} = 14 $$\n\n6. **Calculate the angle measure $\angle GKL = (8x + 11)^\circ$:**\n$$ 8(14) + 11 = 112 + 11 = 123 $$\n\n**Final answer:**\n$$ m\angle GKL = 123^\circ $$