Use a flip-flop to detect you're inside the correct sequence. Accumulate the sequence in a variable, use matching in scalarised list context to get the number of occurrences (the Goatse/Saturn operator). If the patterns can overlap, use a look-ahead instead of a plain match.

#! /usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my $seq;
while (<DATA>) {
    if (my $line_number = (/^>JJ22$/ ... /^>/)) {
        chomp;
        $seq .= $_ unless $line_number == 1 || $line_number =~ /E0/;
    }
}
my $count = () = $seq =~ /[MT].[KA]/g;
say $count;

__DATA__
>JJ57
MKIKLVTVGDAKEEYLIQGINEYLKRLNSYAKRETIEVPDEKAPEKLSDAEMLQVKEKEGEYVFVLAI
NGKQLSSEEFSKEIFQTGISGKSNLTFTTCFSLGLSDSVLQRIMKGEPYHKL
>JJ22
MDQNGASGSHPNRASTRKGAHARERGATVSAMSANRSNIIDEMAKICEADRQTFAIARRTRNESQ
FFGFRTASNKAIEITEAMEKRGAMFLTQSKATDQLNGWQPSDEPDKTSAESEPWFRGKQLSSEEFS
KEIFQTGISGKSNLTFTTCFSLGLSDSVLQRIMKGEPYHKL
>JJ41
MWKTVAPIFAAIFAVGLCGTFRTNTRKGEPTTKCFVFVHDTKARIYQCTFKTWSCPWLNNIVSAQF
QFVTGANYKIVVKLVGELFTETALFNWSSPTTIFTGLGTLITADKTLDCDSNML
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The first check in the unless part excludes the header, the E0 is appended to the last line number, so it excludes the next header.