Give the Sum Line Lengths tool a try, from the Vector > Analysis Tools menu:

[![enter image description here][1]][1]

This will create a new shapefile that summarizes each **input polygon vector layer** feature by the length of **input line vector layer**.

The output shapefile should have an attribute field called **LENGTH** as you can see in the above screenshot, but you can change that to SUM_LENGTH or something similar.

Then you can visualize your map on that field, if that is your purpose:

[![enter image description here][2]][2]

**NOTE** the units are important to your inputs, so in my example decimal degrees are not too meaningful...but if your input coordinate system is in metres you should be good.


  [1]: https://i.sstatic.net/H88PP.png
  [2]: https://i.sstatic.net/tqbAk.png