my $start_time = time(); # remember when we started
my $progress;

while (<$input_file>) {
    ...
    if (time - $start_time > 2) { # 2 seconds have passed
        $progress = create_progress_dialog();
        Tk->update(); # I don't know if there is such a thing or whether Tk needs it
    };
};

undef $progress; # hide progress dialog if we have one

##</code><code>##

use Time::HiRes qw(time);

my $progress;
my $popup = Tk->after( 2, sub { $progress = create_progress_dialog } );

my $last_poll = time();
while (<$input_file>) {
    # let Tk process its events every 0.1 seconds:
    if (time - $last_poll > 0.1) {
        Tk->process_one_event();
    };
    ...
};

if ($progress) {
    undef $progress;
};
undef $popup; # cancel our Tk->after() timeout