kn is a modified Bessel function of the third kind, proportional to K{n+1/2). Also Sqrt is coded incorrectly. Here's what I get for numerical solution of your problem:
In[14]:= d = 0.000012401; a = 0.0382;
FindRoot[0.5 + ((2 Sqrt[x])/
0.000012401) (BesselK[3/2, (0.000012401/(2 Sqrt[x]))]/
BesselK[1/2, (0.000012401/(2 Sqrt[x]))]) ==
1/(4*0.0382), {x, .1}]
Out[15]= {x -> 1.55154*10^-10 - 9.43222*10^-21 I}
In[16]:= Chop[%]
Out[16]= {x -> 1.55154*10^-10}
