Well, once I figured out how to work the debug mode, it was all downhill from there. I assume you all had a good laugh about the "I just keep getting a 'DB(n)' Prompt that won't take my input." Oh well, you're only ignorant once. The program is now giving a reasonable approximation of its intended functionality. I'm attaching the code below so everyone can point out the silly looking, rough-hewn parts. Of course, in the words of the immortal Beeblebrox, "Hey! Don't knock it, it worked."

#!/usr/bin/perl -w
#Purpose: To Take a DOS file wildcard and thus take all the matching c
+ustom SGML files in the working directory and subdivide them into new
+ files whose names are the id's of the divs in those original files.

use strict;

print "Enter the name of a file containing the list of files you want 
+to work on.\n";
our $lines = "";
our @InFileNames;
our @OutFileNames;
our @OutFileExtensions;
our @OutFileContent;
my $i = 0;
my $j = 0;
my $k =0;
my $TheFile = <STDIN>;
chomp ($TheFile);

#open file and get all text
sub OpenFile {
open(FILE, $_[0]) or $lines = "";
local $/ = undef;
$lines = <FILE>;
#remove blank lines
$lines =~ s/\n{2}/\n/gms;
close(FILE);
}


#add ¥ to closing div tags
sub MarkClose {
$lines =~ s/(<div type)/¥$1/gms;
$lines =~ s/\A¥//gms;
}

#open output.txt for appending and write results to it
sub FileAppend {
my $Outfile =  ">>" .  $_[0] . "." . $_[1];
my $Content = $_[2];
open(FILE, $Outfile) or die "Can't open $Outfile.\n";
print FILE $Content;
print FILE "\n";
close FILE;
}

#Create an array containing all file in the directory matching the glo
+b.
sub GetInFileList {
my $FileDef = $_[0];
open (FILE, $FileDef) or die "That isn't a valid file, Wesley!";
local $/ = undef;
$lines = <FILE>;
#remove blank lines
$lines =~ s/\n{2}/\n/gms;
close(FILE);
@InFileNames = split /\n/, $lines;
#If the program can't give a list to Muhammed, than Muhammed will give
+ a list to the program.
}

#Populate an array with the contents of the id attribute of every <div
+> tag in the input file.
sub GetOutFilesList {
$k = 0;
@OutFileNames = $lines =~ m/<div type[^>]*>/gms;
while  ($k < (scalar(@OutFileNames))){
    $OutFileNames[$k] =~ s/<div type="[^"]*" id="([^"]*)"[^>]*>/$1/gms
+;
    $OutFileNames[$k] =~ s/\./_/gms;
    $k = $k + 1;
}
$k = 0;
@OutFileExtensions = $lines =~ m/<div type[^>]*>/gms;
while  ($k < (scalar(@OutFileExtensions))){
    $OutFileExtensions[$k] =~ s/<div[1-9]? type="([^"]*)" id="[^"]*"[^
+>]*>/$1/gms;
    $OutFileExtensions[$k] =~ s/\./_/gms;
    $k = $k + 1;
}
}

#Subdivides the File into the subfiles.
sub GetOutFileContent {
my $LinesString = $_[0];
@OutFileContent = split /¥/,  $LinesString;
}

### Does the job
&GetInFileList($TheFile);
$i =0;
while ($i < (scalar(@InFileNames))) {
    &OpenFile($InFileNames[$i]);
    &MarkClose();
    &GetOutFilesList;
    &GetOutFileContent($lines);
    $j=0;
    while ($j < (scalar(@OutFileNames))){
        &FileAppend($OutFileNames[$j], $OutFileExtensions[$j], $OutFil
+eContent[$j]);
        $j = $j + 1;
    }
    $i = $i + 1;
}

#be nice and say it's done
print "Program Finished\n";
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(Edited to reflect the code that actually DID work.)