Let's denote Karen's older sister's height as \( x \) inches. According to the problem, Karen's height is 27 inches more than half of her sister's height. This gives us the equation:
\[ 59 = \frac{x}{2} + 27 \]
To solve for \( x \), first, we can isolate \( \frac{x}{2} \) on one side of the equation:
\[ \frac{x}{2} = 59 - 27 \]
Calculating the right-hand side:
\[ 59 - 27 = 32 \]
So now we have:
\[ \frac{x}{2} = 32 \]
Next, we multiply both sides by 2 to find \( x \):
\[ x = 32 \times 2 = 64 \]
Thus, Karen's older sister is \( \boxed{64} \) inches tall.