The value from the second street line is transmitted to DHL (visible in the request in the log). However, the value is not printed onto the PDF Label.
The DHL webservice offers fewer address fields than Magento. At DHL, there is only a single field for the street. Magento however offers two fields in the checkout by default (or more, depending on the configuration).
The additional data is transmitted as "addressAddition" to DHL, but this data is processed by DHL only for specific destination countries. E.g. for shipments to Austria or Italy the value is printed onto the label, but not for shipments to Germany.
This behavior of the DHL Business Customer Shipping API cannot be changed through the Magento extension.
To make sure that additional data is always visible on the label, the Magento checkout field "Company" can be used instead of a second street address line. The value is mapped at DHL to the field "Name2" and is always processed. There is no validation, which means the customer can enter arbitrary data.
Recommendation: In the checkout template, change the label for the field "Company" to show that other information can be entered as well (floor number, apartment, gate, entrance, etc.)