Research, especially in the social sciences and humanities, is increasingly reliant on the application of data science methods to analyze large amounts of (often private) data. Secure data enclaves provide a solution for managing and analyzing private data. However, such enclaves do not readily support discovery science---a form of exploratory or interactive analysis by which researchers execute a range of (sometimes large) analyses in an iterative and collaborative manner. The batch computing model offered by many data enclaves is well suited to executing large compute tasks; however it is far from ideal for day-to-day discovery science. As researchers must submit jobs to queues and wait for results, the high latencies inherent in queue-based, batch computing systems hinder interactive analysis. In this paper we describe how we have augmented the Cloud Kotta secure data enclave to support collaborative and interactive analysis of sensitive data. Our model uses Jupyter notebooks as a flexible analysis environment and Python language constructs to support the execution of arbitrary functions on private data within this secure framework.