1
Julia 0.4, 44 bytes
f(x,k=endof(x)+1)=(try k=2find(x)[k/2]end;k)
This function accepts a single integer in form of the list of its binary digits as input.
The algorithm is equivalent to the one from @Agawa001's answer and my Jelly answer.
Try it online!
How it works
find(x) returns the 1-b...