Splitting a graph into minimum number of subpaths
Your problem (as stated) seems to be NP-hard.  Here is a reduction from Partition.  

Given an instance of Partition (a collection $x_1,\ldots,x_n$ of positive integers), construct a graph with $n+1$ vertices $v_1,\ldots,v_{n+1}$ and, for each $i$, two edges from $v_i$ to $v_{i+1}$: one of cost zero, one of cost $x_i$.  

Suppose the Partition instance is satisfiable. That is, there is a subset $S$ of $[n]$ such that $\sum_{i\in S} x_i = \sum_{i} x_i/2$.  Then there are two paths that together cover all the edges, with each path costing $L=\sum_{i} x_i/2$.  

Conversely, if there are two paths each of cost at most $L$ (that cover all the edges), the two paths together must cover each $x_i$ exactly once, and the total weight on each must be exactly $L$. So there must be a subset $S$ such that $\sum_{i\in S} x_i = \sum_i x_i/2$.  That is, the Partition instance is satisfiable.

Perhaps you have further restrictions (e.g. polynomially bounded integer edge weights)?