Based on the description of your requirement, it sounds like you will need to fork two children - one to check the server, one to do the countdown. Something like this:

use strict;
use warnings;
use feature qw(:5.10);

# Fork off two child processes
my ($who_am_i, $countdown_pid, $checking_pid);
$countdown_pid = fork();
if ($countdown_pid == 0){
   $who_am_i = "countdown process";
} elsif ($countdown_pid){
   $checking_pid = fork();
   if ($checking_pid == 0){
      $who_am_i = "checking process";
   } elsif ($checking_pid){
      $who_am_i = "controlling process";
   } else{
      die "Error forking: $!";
   }
} else{
   die "Error forking: $!";
}

# Each process does different things:
given($who_am_i){
   when ("countdown process"){
      sleep_count(3, 'Verifying with Nagios that resin is down: ');
   }
   when ("checking process"){
      # repeatedly do server check via "nagios"
   }
   when ("controlling process"){
      # Wait on the "checking process" to complete.
      waitpid($checking_pid, 0);
      # Kill countdown timer
      kill("TERM", $countdown_pid);
      waitpid($countdown_pid, 0);
   }
}
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You should probably check the exit status of your child processes (via $? after the waitpid calls).

Hope this helps.

Update: Actually, there's no need to have three processes. Two is enough, as in your post, although conceptually I like the idea of delegating each subtask to separate child processes.

Perhaps the only piece that was really missing from your code was killing off the countdown timer.