First, reduce this fraction into $\frac{13}{476}$. Note that $476 = 119\cdot 4$ and hence the binary expansion of $\frac{13}{476}$ should be exactly the same as $\frac{13}{119}$ except it is shifted two digits rightwards. Then consider the form given in the question, expand it and we have the sum of two terms, one of which is the sum from the non-repeating part and another is from the repeating part. $119$ should divide the denominator, or the RHS would not be an integer when two sides are multiplied by $119$. Now $119 = 7 \cdot 17$, and it suffices to find $k$ such that $2^k \equiv 1 \pmod{117}$. Now use Fermat-Euler Theorem to find the order of $2$ modulo $117$, which should be exactly $24$.