Upvote:6
Contact the company.
If they don't reimburse you, contact your credit card bank, and revoke the incorrect payment; simple tell them the service you bought was not provided, and the provider is not working with you.
Note that middle names are irrelevant on flight tickets, so simply leave them off in the future. If you never cancelled the first ticket, you might need to able to prove that they told you to proceed that way.
Generally, 'cheapo' websites are typically cheap because they have little or unqualified customer support, and leave you alone when there is an issue. I recommend to always buy from the airline directly.