If you have any books on basic programming, look up recursion. This is the kind of task that computer science profs make up to debate iterative versus recursive programming techniques.

To solve this just using loops (iterative technique), you have to build a stack of some sort to keep track of how many levels of braces you've processed. Fortunately Perl makes this trivial by using perlfunc:push and perlfunc:pop.

For a recursive technique, take your current while loop and make it a sub. Then call the sub again when you encounter an open brace. Exit the sub when you match the closing brace, and print along the way. You're still using a stack, but it's the call frame stack instead of an array you manipulate yourself.