Before answering that question, let's start by asking a basic question: what enables drug lords to have power in the first place? To repeat an oft-used cliché, money is power. The reason they can literally get away with murder is because they have a constant stream of drug money coming in, most notably from the United States. We can talk about addressing judicial reform, police reform, the ability to enforce laws (I think fighting a war against impunity would be a good idea, but tough to implement), providing Mexico with foreign aid to fight the War on Drugs, anti-laundering restrictions (which don't help as much as one would think), or the ever-amorphous anti-corruption reform to no avail. Mexico's democratic institutions are most probably too weak to handle the cartels on their own. Until you cut off the cartels' life support, i.e., money, they will continue to be supported by the demand in the underground market.
The Cato Institute, Instituto Mexicano para la Competividad, and the Center for Strategic and International Studies (p. 42) came up with a bright idea: legalize marijuana, which would be an important step in ending the War on Drugs. Even with the three states that have legalized it in the United States, it seems to have some effect already. According to the U.S. Department of Health and Human Services survey data, the vast majority of illicit drug usage in this country is marijuana-based. Legalizing marijuana would, amongst other things, close the gateway that exposes many to the harder, illicit substances.
Last year, five Nobel Prize-winning economists published a report through the London School of Economics (LSE), and they found that the War on Drugs has been a waste of time and money. Why?
In economic terms, implementing a supply-reduction intervention like the War on Drugs backfires. Normally, a supply reduction would cause a price increase, which would deter would-be offenders. However, with illicit drugs, the price increase incentivizes new supply by shifting commodity supply chains (LSE, p. 10). This returns us to a point [E2] similar to the market equilibrium prior to the War on Drugs. The elasticity of demand for the illicit drugs causes the price increase to be much larger than the quantity decrease (see Figure above). The black-market premium is what keeps the drug lords rolling in money. Decriminalization, or even better, legalization, would hit the drug lords where it hurts: their wallets. This clearly won't happen overnight. The federalist system in the United States, in which each state would have to approve legalization before putting enough pressure on the federal government, will take time. Even though it is moving at seemingly glacial speeds, other countries can follow suit. As much as I think that having stronger governmental institutions and less corruption than it currently has would help immensely, I would opine that marijuana legalization would help the Mexican government put a quash on drug trafficking sooner than institutional reform.