Perhaps use Simplify to handle the denominators.

In[1]:= Clear[m, k, l, a, b, A, B];
simp = Simplify[1/Sum[Binomial[m-k, l]*(-1)^l*Gamma[k+a]/(        b + B + k/A^2)^(k+a), {l, 0, m-k}]*
                  Sum[Binomial[m-k, l]*(-1)^l*Gamma[k+a]/(1/L^2 + b + B + k/A^2)^(k+a), {l, 0, m-k}]];
m = 35; k = 32; \[Lambda] = 1; a = 2; b = 2; u = RandomReal[UniformDistribution[{0, 1}], m];
t = ((-1*\[Lambda])/Log[u])^(1/2); s = t; w = Sort[s]; st = Take[w, k]; B = Total[st^-2]; A = Last[st];
FindRoot[simp == 0.25, {L, 1}]
FindRoot[simp == 0.75, {L, 1}]
Plot[{simp - 0.25, simp - 0.75}, {L, 0, 3}]

Out[4]= {L -> 0.72545}

Out[5]= {L -> 1.60547}

Please check this carefully. Thank you