The constitutional obligation to provide an attorney for indigent defendants falls to the jurisdiction that has filed the action in its courts. If you're arrested for a municipal violation, Denver provides a municipal public defender. The federal government doesn't provide you a lawyer if you're in Denver municipal court. Similarly, if you are arrested on a state criminal charge, the state will provide a state public defender if you are indigent and can't afford one, not Denver.

So if federal agents arrest you for being an illegal immigrant and bring you before a federal immigration court, then the feds should provide the lawyer. The city has no business spending city taxes to provide legal defense for something that is not a city charge. This is entirely inappropriate.