Last night I came across an interesting Torah of Rebbe Nachman's in the Likkutei Torah on Tefillin. Rebbe Natan mentions in passing that Erech Apayim, (ארך אפים) being slow to anger, is the root of all the 13 attributes of mercy. This is hinted at by the fact that Erech Apayim means long-faced. How is a face lengthened? through a beard. The thirteen attributes of mercy are rooted in the thirteen tikkunim of the beard.
We're getting into areas that seem so confusing and tangled as to make us throw up our arms and say we don't know or understand anything. What does the beard have to do with not getting angry? What do the tikkunim of the beard even mean? etc. I could start to try and explain the little that I know, but actually in the same Torah, Rebbe Natan explains that getting to this place of newness, of recognising that we don't know anything (and serving HaShem from that place) is the key to being old and young at once.
Not satisfied? Getting a little annoyed or even angry? I'll try and explain a little instead of what seems to be talking in circles. The beard, (as well as all hair on the body) actually represents shefah, the potential for the emanation of divine light that is hidden within us. The thirteen different kinds of emanation from the face downwards to all that lies below are the thirteen tikkunim of the beard. These thirteen emanations, when they come down from the highest world are the thirteen attributes of mercy. They are comprised of the ten sephiroth of the particular world in question in addition to the three sephiroth from the world above which protrude down into the world below, Netzah Hod and Yesod.
Normally according to the Holy Ariz"l, Notzer Hesed is the central attribute upon which all the other 12 rely. It is interesting then that Rebbe Nachman explains that Erech Apayim is the root attribute of the 13 attributes. Especially since Erech Apayim is sometimes counted as two of the attributes and not one. So, the best answer I have to explain Rebbe Nachman is this: The source of the 13 attributes is in Keter, the highest sefirah, the lower half of this Sefirah is called (in certain contexts) Arich Anpin, the long face. Erech Apayim means essentially the same thing, just in hebrew. So, Rebbe Nachman is referring to the root/source of the 13 attributes and isn't referring to one of the attributes specifically.
The other more practical explanation is that Rebbe Nachman is teaching us the way to draw down the thirteen attributes of mercy: through patience and strength in the face of anger, we are able to draw down HaShem's thirteen attributes of mercy into this world. In this way Erech Apayim is the root of HaShem's thirteen attributes of mercy in this world.
So, while this doesn't really give us any tips as to how to work on Erech Apayim, it does perhaps explain a little of why it is so difficult to accomplish and why the yetzer hara fights it so hard.