On the topology toolbar is the tool Planarize Lines which can be used in Standard and Advanced license levels and then Explode Multipart features on the Advanced Editing Toolbar, the lines do not need to participate in a topology to be subject to this tool.
Start editing, select all and activate the tool, enter a suitable tolerance and done! Don't forget to save your edits.
A tool that also may help is Multipart to Single Part which will break the lines down to single part features, which is probably what the tool is having issues with: each feature should be a single part geometry.
If you are in a situation where one line is continuous and the other terminates without an intersection you can use Feature Vertices to Point on the lines with an option of both ends to generate end points. Use the generated end points and original lines in Split Line at Point tool which will break the continuous line allowing for a true topological intersection. Follow that with an Integrate to ensure ends are confluent, then Repair Geometry with delete empty to remove very short segments and then just to be sure Multipart to Single Part.