Be forewarned, I don't know if this might cause other issues such as shorting out your printer or if it's going to work for all inks. This was HP with OEM cartridges that had the print head on the cartridge.
I don't do much printing so clogged heads is common problem, while on the topic be sure to turn your printer off while not in use as this puts it in storage mode which will help prevent clogged heads. Since printer ink costs thousands of dollars per gallon using it to clean the heads can get quite expensive if you don't print that much .
If you look at your print cartridge if the print head is right on the cartridge there will be a little metal strip on the underside and if you look really close you'll see holes. These holes are what get clogged. In the past I've often used isopropyl alcohol, just get a q-tip a wet with the alcohol and run it over the metal strip. Make sure it's dry and then put it back in the printer, worked like a charm every time and no wasted ink on cleaning it.
This time however I had two cartridges that were sitting for a year and a half, one cartridge was almost brand new. The print out wasn't just screwed up, there was no printout. In this case the trick with the q-tip produced some results but far from anything I could use and cleaning it with the ink didn't do anything either. . The next step was I put about a quarter of an inch of alcohol in some ramekins and set the cartridges print head down in it. I let them soak for about 6 hours and then let them dry for about 1 hour after cleaning up most of the alcohol. That worked a little better but still not good enough even for printing where quality isn't that important.
After that failed I came up with a better solution, having nothing to lose (other than maybe a printer). I put about 1/4 inch of water in the ramekin, brought the water to a boil in the microwave and then set the cartridges in it with the print head down of course. Almost immediately I could see ink seeping into the water, I let it go for an hour and when I got back the water was jet black. I had originally intended on leaving them in there much longer but I think I may have actually had them in there too long and possibly lost some of that three thousand dollar a gallon ink. I took them out and and blew some air into the crevices to try and make sure everything was dry, I also tapped them on a newspaper to knock any droplets loose if there was any. I let them sit for two hours and didn't see any water anywhere. Popped them in and It worked perfectly, it was like a brand new cartridge.
I guess the test now is if it works tomorrow, I'll let you know.