Streets that are expected to connect have to have either a) and endpoint or b) a vertex on both street features at the location where they touch, depending on whether your connectivity policy is End Point or Any Vertex. Please read this doc page to understand what those mean: ArcGIS Help (10.2, 10.2.1, and 10.2.2)
If your connectivity is Any Vertex and you just need to do a batch vertex insertion at locations where your streets touch, you can run the Integrate tool. Make sure you make a back-up copy of your data first, as Integrate actually modifies the input features by moving them slightly. You can set the tolerance to be small or tiny, though, so it won't move stuff much or at all and will instead just add vertices. The one problem here is locations where you have an overpass and the crossing streets are actually not supposed to connect. You might have to manually fix those areas.
If your connectivity is End Point, you may have more trouble. If you split existing features down the middle so that they have end points, you risk messing up attributes that need to be carefully apportioned when the street feature gets split. For instance, if you have a field called "walktime" and you split a street that has a value of 5 minutes in the walktime field, it would be incorrect for the two split parts of the street to both have a value of 5 minutes. So, if you have End Point connectivity, you either need to split a lot of features very carefully, or you need to switch to Any Vertex connectivity and run the Integrate tool.