This is quick, ugly, and prone to fail at the slightest change in the data, but maybe it'll get you started:

open INPUT,'<',$out_file or die "Can't open file " . $out_file . "\n$!
+\n"; #Open for read
    while (<INPUT>)
    {
        my $name = $connect->quote($_);
        my $address1 = $connect->quote(<INPUT>);
        my $address2 = $connect->quote(<INPUT>);
        my $phone = $connect->quote(<INPUT>);
        for (<INPUT>) {
            next if /(Council|^Continuing|^Mapping)/;
            last;
        }
        my $overall = $connect->quote($_);
        my $inspections = $connect->quote(<INPUT>);
        my $staffing = $connect->quote(<INPUT>);
        my $quality = $connect->quote(<INPUT>);
        my $programs = $connect->quote(<INPUT>);
        my $beds = $connect->quote(<INPUT>);
        my $ownership = $connect->quote(<INPUT>);
        my $query_string = "INSERT INTO nursing_homes (name, address1,
+ address2, phone, overall, inspections, staffing, quality, programs, 
+beds, ownership) VALUES ($name, $address1, $address2, $phone, $overal
+l, $inspections, $staffing, $quality, $programs, $beds, $ownership)";
        #printVariables($name, $address1, $address2, $phone, $overall,
+ $inspections, $staffing, $quality, $programs, $beds, $ownership, $qu
+ery_string);
        my $query_handle = $connect->prepare("INSERT INTO nursing_home
+s (name, address1, address2, phone, overall, inspections, staffing, q
+uality, programs, beds, ownership) VALUES ($name, $address1, $address
+2, $phone, $overall, $inspections, $staffing, $quality, $programs, $b
+eds, $ownership)");
        $query_handle->execute();
    }
[download]

You define $query_string but don't use it, and you're not checking return values for the prepare and execute calls. You must.