It's a slightly tricky match to get right depending on just how fussy you really want to be. Using a negative look ahead helps cover most bases though. Consider:

use strict;
use warnings;

my $str = <<STR;
Errors: 0 failed
Errors: 01 failed
Errors: 1 failed
Errors: 10 failed
Errors: None failed
Ben10 failed
STR

open my $fIn, '<', \$str;

while (<$fIn>) {
    print if /\b(?!0+\b)\d{1,9} failed/;
}
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Prints:

Errors: 01 failed
Errors: 1 failed
Errors: 10 failed
[download]

The (?!0+\b) is the negative look ahead that fails the match if a zero value (regardless of the number of 0 digits) is found. The \b's are there to ensure all the digits in the number are matched.