They’re essentially their own governments in large portions of Mexico, and significantly more powerful than the federal government.
There’s no real solid answer to it, but to eliminate the cartels will require massive collaboration both politically and militarily between the US and Mexico. Which likely won’t happen. Mexico would never agree to allow a deployment of US troops even if working hand in hand with the Mexican military, and the US government would never put US troops under foreign command.
But it’s going to have to take a much more powerful military to eradicate the cartels, who themselves are pretty powerful militaries.