Like so many things, the solution is easy to understand once you know it but difficult to see when you don't know where to look.

Your program reads the last two lines and copies them into your variable $line but it exits without ever printing the contents of this variable. All you have to do is print the contents of this variable after reading the last line of the file.

Your loop terminates after reading the last line of the file. Therefore, to print the contents of $line after reading the last line of the file, you need to add a print statement after your loop.

Your program is otherwise a bit more complex than it needs to be. Here is a simplified version, with a final print added, to print that troublesome last line.

#!/usr/bin/perl -w
use strict;
use warnings;

my $line;

while (<DATA>) {
    chomp;

    if (/^$/) {
        print "\n";
        $line = "";
    } elsif (/^[A-Z]{2}:\s/) {
        print "$line\n" if($line);
        $line     = $_;
    } else {
        $line .= " $_";
    }
}

print "$line\n" if($line);

__DATA__
DN: Data Name
Information Sciences
TI: Title
Distribution and abundance of bound volumes of the file in institute
shelves from 1978
AU: Author
Sahu, SR; Gawas, AKG
AF: Affiliation
Inst. of document scanning and documentation profiling at library,
toward the prograssive society and establishment
SO: Source
Test document file. Vol. 5, no. 1, pp. 1-14. Jul 1990.
DE: Descriptors
Article Subject Terms: Abundance; Ecological distribution;
Geographical distribution; Life cycle; Zooplankton; Article Taxonomic
Terms: Euphausia; Article Geographic Terms: ISW,
LA: Language
English
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