An almost nonexistent Triangle

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This problem is a straight application of Heron's formula, which gives the area of any triangle, given the sides. It follows:

$\text{Area } = \sqrt{s(s-a)(s-b)(s-c)}, s = \dfrac{a+b+c}{2}. \text{ Here, } s = \dfrac{15+11+\frac{77}{3}}{2} = \dfrac{155}{6}\\ \implies \text{Area } = \sqrt{\left(\dfrac{155}{6}\right)\left(\dfrac{65}{6}\right)\left(\dfrac{89}{6}\right)\left(\dfrac{1}{6}\right)}\\ = \sqrt{\left(\dfrac{5\cdot 5}{36\cdot 36}\right)\left(31\right)\left(13\right)\left(89\right)}\\ \implies \text{Area } = \dfrac{5}{36}\sqrt{(89)(31)(13)} \\ \therefore r+s+k+m+n = 5+36+89+31+13 = \boxed{174.}$