For whatever reason, WordPress could not remove all/part of the ‘old’ version of Event Espresso and so the update failed.
Can you see the core Event Espresso plugin listed in Dashboard -> Plugins at all?
(Note, not the add-ons, the main plugin itself, it should show as ‘Event Espresso’ with no additional text in the name)
If so, delete that version from your site (your event/registration data will not be lost) and then download the latest version from your account page and upload that on your site in Dashboard -> Plugins -> Add new -> Upoad plugin.
Activate and you’ll now have the latest version.
If you try to install the above and WordPress tells you the destination already exists, you’ll need FTP access to the server so you can manually remove the /wp-content/plugins/event-espresso-core-reg/ directory from your site and then repeat.