I think your logic is slightly amiss when you calculate the "start of this day". This variant works for me:

#!/usr/bin/perl -Tw

use strict;

print "Today: " . hours_in_day (time) . "hours.\n";
print "Sun: " . hours_in_day (time-170000) . "hours.\n";

sub hours_in_day {
    my $now = 60*60*int( shift()/60/60 );   # Start of this hour
    my $hour = ( localtime($now) )[2];
    $now -= $hour*60*60;                    # Offset from start of thi
+s day
    $hour = ( localtime($now) )[2];
    $hour += 24 unless int($hour/12);
    return $hour;
}
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and gives this output for me (Sunday was the switch from BST to GMT here):

Today: 24hours.
Sun: 25hours.
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