If you are relying on a numeric field called something like Miles rather than the Shape_Length, Length, etc. field maintained by the feature class in a Geodatabase, then you will have to manually recalculate the Miles field. You can use the Geometry calculator to update the field based on the geometry Property of Length and set the Units to Miles.
Setting up a split policy to proportion the value of the field relative to the length may work with the Split Line at Vertice tool, but it may not, since not all geoprocessing tools respect the split policy.
For the maintenance of the data, you should use Attribute Assistant. It can be set up to automatically update attributes during an edit session in response to feature creation, geometry changes, or attribute changes. For example, using the Expression method it can automatically update your Miles field every time a feature is created or has its geometry modified using the formula: [Length] / 5280 (assuming your native units are in feet). Attribute Assistant can also be set up to automatically update a Minutes field every time a feature gets created, the geometry changes or the speed limit attribute changes. The Expression for minutes when length is in feet and speed is in MPH is: ([Length] / 5280) / ([Speed_MPH] / 60). Attribute Assistant is invaluable for editing feature classes with lots of attributes that need to maintain a relationship to the feature geometry or between interdependent attributes.
However, you do not need a field called Miles for NA to report your distances in Miles, even if your native units are in feet. By default NA should detect and use the geodatabase Length field for cost. As part of the configuration of the Network Dataset you specify the units you want NA to report the native feature class length maintained by the geodatabase and can specify Miles as the units. The way you do that is dependent on the ArcGIS version of NA that you are using, so specify the version in your original post or in a response.
I also highly recommend setting up a topology for your centerlines to detect dangles and lines that cross each other. The topology makes it easy to detect these types of errors in your connectivity and easily correct them by doing splits, extends, snaps, or trims using the topology toolbar tools. Even if you don't use a geodatabase topology, the topology toolbar has a Planarize tool that will split selected lines during an edit session where they intersect each other, whether or not a vertice exists at the intersection.