I know an email was sent using the online service emkei.cz. Is there a way to find out who used that service to send the specific email I received?
This is a seriously broad question and probably off-topic.
Still, emkei.cz is a free online mailer. It does not require an account or any form of registration. This means that any information contained in the email itself will be useless when it comes to identifying the ultimate sender.
The legal way for authorities to identify would be to subpoena the service provider itself. If you notice the small print at the bottom, they say:
This service does not violate the EU law. We are not obliged to keep any logs. FinalTek.com and Forpsi.com are neither owners of this service nor responsible for its content.
At a guess, I would say they might not be logging source IPs of email senders, so even a subpoena would be fruitless.
Of course, those are the normal and legal ways to find out who used the service. Help with any alternate methods is definitely off-topic here.