You could save some effort by reading in paragraph mode rather than line-by-line.

#! perl -slw
use strict;

local $/ = '';
my $n = 0;
while (<DATA>) {
    $n++;
    next unless /INFORMATION I AM LOOKING FOR 2/;
    print 'Found the info at record ', $n;
    print;
}

__DATA__

START OF RECORD
  LINE 1
  INFORMATION I AM LOOKING FOR 1
  LINE 3
  LINE 4
  LINE 5
  LINE 6
END OF RECORD

START OF RECORD
  LINE 1
  LINE 2
  LINE 3
  LINE 4
  INFORMATION I AM LOOKING FOR 2
  LINE 6
END OF RECORD
[download]

Gives

C:\test>227789
Found the info at record 2
START OF RECORD
  LINE 1
  LINE 2
  LINE 3
  LINE 4
  INFORMATION I AM LOOKING FOR 2
  LINE 6
END OF RECORD

C:\test>
[download]

Alternatively, setting $/ = 'INFORMATION I AM LOOKING FOR' then <read>ing once will get to the position of the end of that info. You then use seek to back up the read pointer by the size of a record (assuming they are consistant size) and then set $/ = '' and <read> twice. The second <read> should be your record.

If you records are of wildly varing sizes you would have to backup by the size of the largest record and then <read> forward line-by-line ($/="\n") until you get a blank one, then set $/= '' and <read> should get you the whole record.