The only time I can imagine a scenario in which an Enterprise would purchase software within a a one week period would be a solution to a critical bleeding pain: A security breach, a pending SEC violations, a BI solution which can confirm/deny internal nefarious activities, operational software for managing during a flood/whiteout/radiation leak.
You get the idea.
Enterprises companies purchasing takes a long time because -- well, there are lots of reasons and that wasn't really the question -- but let's just say there are lots of them. Some are entirely reasonable and understood. Others, well not so much. No matter what the reasons, they are there and very difficult to "end run."
Absent a "bleeding pain" solution -- I believe the up-sell from a personal solution to an enterprise level solution as mentioned by @richardg and @elie is your best bet in this innovative and creative marketing/sales strategy.
Creating an individual version of the critical function of your application and make it usable within a corporate firewall -- ideally so the user will not even need to secure permission from the IT gatekeepers. Become essential-- and let them be your advocates for the sale of an enterprise version.