I have the raster below which I want to convert to a set of vector lines:

As you can see, there are small breaks in the lines and some bits are double-width. I want to be able to reliably import these sort of images to a vector in ArcGIS.
I've spent a lot of time trying to get tools like Integrate, Extend Lines, Trim Lines and Unsplit Lines to do what I want, but I just can't seem to achieve it. I guess this problem must have been experienced before, does anyone have any ideas for a workflow that would do what I want?
I thought of maybe trying to clean up the raster before-hand, but I'm not sure how to go about that. Again, any ideas would be much appreciated.