“We expect this to problem to continue and there is no short-term solution in sight," said Osorio Chong during a visit to a refugee camp in Mexicali.
The drug lord will remain in ICE detention while the agency makes arrangements to carry out his removal to Mexico.
Four other suspects including a cartel commander have pleaded guilty in the same cases. They were among five extradited over the affair.
A former ICE attorney pleaded guilty Friday after prosecutors said he forged a document to make it look like a Mexican citizen who wanted to stay in the U.S. was not eligible to do so.
El IPM registró un aumento mensual de 0.11 puntos, al ubicarse en 52.1 puntos
The Board of Supervisors unanimously approved a resolution that urges the sheriff not to participate in a detainer-notification system that asks jails to let Immigration Customs and Enforcement officials know when an inmate of interest is being released.
The gradual implementation of the new rules started last June and is expected to be concluded by January 2016.
Nearly eight of 10 arrested were from Mexico, with the rest from 20 other countries.
Diego Oswaldo Jauregui Rosales is wanted for a fatal Christmas shooting in Guadalajara four years ago.
ICE says that two lawyers violated visitation standards, but a coalition of immigration attorneys says the bans are unprecedented.