I don't know of a plugin that allows you to do this out of the box, but as you are (/seem to be) comfortable with database manipulation, I've outline below how you can achieve what you want using custom coding.
You can run custom queries using the wpdb-class. There are a few posts on WPSE/SO that treat this. Once you've set those up, you can run them on, ...
You're missing your where statement. You need to tell wpdb which row it should be updating, otherwise you're looking to insert rather than update.
From the $wpdb::update Codex:
wpdb->update( string $table, array $data, array $where, array|string $format = null, array|string $where_format = null );
Add this to functions.php
and then make either of following changes in your code.
Because your jquery code is sending FirstNM to server (not FirstName).
Therefore in post_language() function, change this line
$FirstName = isset( $_POST['FirstName'] ) ? $_POST['FirstName'] : '';