How do zen practitioners balance the mix between Buddhism detachment and Taoism attachment?
To say that Zen = Buddhism + Taoism is an utter simplification. To say that Buddhism seeks "detachment from all existence" is not at all accurate and to say that Taoism seeks "attachment to all existence" is a caricature.

And yet I will answer this question on its own terms, to help you clear your doubts:

If you detach from all existence, completely 100%, at the end there won't be "you". If you unite with all existence, completely 100%, at the end there won't be "you". In both cases the result is the same.

That said, you should study the Goal of Buddhism (Nirvana or Enlightenment) and the Goal of Taoism (living in harmony with the Tao) to understand what both of them really mean, so you can go beyond simplistic definitions and understand what this is actually about.