How platelets survive without nucleus?

Basically, there's no use for nucleus in a platelet cell, and so there isn't one.

Here's a more thorough explanation:

Platelets are cells that don't go through mitosis as fully specialized cells, thus having a nucleus, with the genetic information for duplication, would be unnecessary, and in nature unnecessary things are often gotten rid of. Another reason is the fact that platelets don't synthesize proteins, they contain substances that participate in the clogging of blood, but they don't produce these substances. When more is needed, more platelets are produced in the bone-marrow.