I just cleaned a very clogged i850 printhead which should be the same as for the i560.
First if you have the printhead out of the carrier, try drops of alcohol on the printhead.
If it passes through the printhead then you're done.
I'd then soak the thing in Windex and clean it with hot water, then try drops on the ink inlet
of the printhead.
If then you need to pull the printhead apart, be gentle. Start by prying the end opposite the
printed circuit flat cable. You will be stopped by the cable end. Gently lift the cable end
while continuing to pry off the non-cable end. Think of lifting the printhead off while trying
to keep it parallel.
once you have it off, you can pull off the plastic ink channel that's clogging your particular
color and soak the whole thing in Windex. You'll see the clog dissolve in the Windex.
Took several soaks to get it all clean for me. now I have black.